Name.

The City is mentioned in ancient sources as "Lissos" (Ancient Greek: Λισσός) and "Lissus" (Latin: "Lissus"). The ancient name "Lissus" evolved into its modern form "Lezhë" (archaic: "Lesh") through Albanian sound changes. In Turkish, the town is known as "Leş" or "Eşim" and in Italian as "Alessio". Lezhë is also known as "Alise", "Alexiensis", "Eschenderari", or "Mrtav".

History.

Prehistory.

From the early Mycenaean period (1600-1450 BC) a free exchange pattern is confirmed with the centres of Mycenaean Greece as seen by various swords (C and D type) unearthed in Lezhë. The earliest human constructions have an Illyrian character and appear on the site from the Late Bronze Age and Early Iron Age. The settlement with its fortifications was built on a 413 m high mountain, the "Mal i Shëlbuemit", from at least the 8th century BC, and was located near the mouth of the Drin river.

Classical antiquity.

In antiquity the area was described as the territory of the Illyrii tribe (the "Illyrians proper";, "Illyrioi"; or "Illyrii propriae dicti").  Diodorus ("Library", 15.1, 1st century BC) mentions that Dionysius of Syracuse founded a "city named "Lissus"" in the year 385 BC, as part of a strategy by Dionysius to secure Syracusan trade routes along the Adriatic. Diodorus calls it a polis. According to some scholars, the Syracusan colony mentioned by Diodorus was established at Issa near the island of Pharos, not at Lissus (modern Lezhë) which was too distant for the events described by the ancient historian. According to Gocha Tsetskhladze, there is nothing to connect Lissos with Syracuse except Diodorus' account, and even if Diodorus' account is accepted as accurate, it is very likely that this colony had a short life.  The earliest of the fortification walls of the proto-urban settlement are of typical Illyrian construction and are dated to the late 4th century BC. The transition from the Iron Age fortification of Acrolissus (on the 413 m "Shëlbuem" mountain) to the proper Illyrian city of Lissus was continuous. The city was built on a lower hill (172 m) near the Iron Age fortification. It was surrounded by ramparts that faced the low valley of the Drin river and the sea coast. Its function was to guard the route inland, to ensure defense against possible attacks from the sea, and to furnish a secure anchorage for the Illyrian ships. By the 3rd century BC, Lissus was one of the main cities of the Illyrian kingdom under the Ardiaean and Labeatan dynasties. In the 228 BC peace treaty with Rome, the Illyrian queen Teuta promised not to sail south of Lissus at the mouth of the Drin river with more than two "lembi" (Illyrian light ships), even those had to be unarmed. But when Rome was engaged in a war against the Celtic peoples of the Po Valley in northern Italy about the years 225–222 BC, Illyrian commander Demetrius detached the Atintani tribe from their alliance with Rome. Moreover, he sailed south of Lissus and engaged in piracy in violation of the 228 BC peace treaty. In the summer of 221 BC, tensions in Greece increased as Macedonia allied with the Achaean League against the Aetolian League, and the Illyrians attacked in their typical manner. Demetrius and Scerdilaidas sailed with 90 "lembi" south of Lissus. When they failed an assault on Pylos (western Peloponnese), they separated their fleets and Scerdilaidas returned north with 40 ships, while Demetrius plundered the Cyclades with 50 ships.  In Roman times Lissus was located in a territory inhabited by the Labeatae, however ancient sources never relate it with this tribe. Taking in account archaeological and historical considerations, the city of Lissus should have been founded in a Labeatan context, but perhaps by the time of Teuta's fall in the end of the 3rd century BC, it was organized as a proper "polis" turning away from it's ethnic context. The dissociation from the ethnic to the "polis" coincided with Philip V of Macedon's conquest of a number of cities in Illyria. In 211 BC, Philip V captured Acrolissus, the citadel of Lissus, and Lissos surrendered to him, becoming the Macedonian outlet to the Adriatic sea. The town was later recovered by the Illyrians. It was in Lissus that Perseus of Macedon negotiated an alliance against Rome with the Illyrian king Gentius, and it was from Lissus that Gentius organized his army against the Romans. Lissus maintained a large degree of municipal autonomy under both Macedonian and Illyrian rule, as evidenced by the coins minted there. During the reign of Gentius in the first half of the 2nd century BC, Lissus minted coins for the Illyrian ruler. The city was of some importance in the Roman Civil War, being taken by Marc Antony and then remaining loyal to Caesar. In Roman times, the city was part of the province of Epirus Nova,

Middle Ages and Ottoman period.

During the reign of Justinian I (527-565) the local fortress was possibly mentioned as "Alistion" in the Synecdemus of Hierocles. At early 590s Lissus was captured by Slavic populations. Byzantine control was re-established during the early 9th century. In Middle Ages Lezha (known in Italian as Alessio) frequently changed masters until the Venetians took possession of it in 1386. It still belonged to them when Skanderbeg died, but In 1478 it fell into the hands of Turks during the siege of Shkodra, with the exception of a short period (1501–1506) when it returned to Venetian domination. Because it was under the Venetian control, it was chosen in 1444 by George Castrioti (Skanderbeg) as a neutral place for the convention of Albanian, Serbian, Dalmatian and other lords of the area aiming at organizing their common defence against the Turks.  Lezha was the site of the League of Lezhë where Skanderbeg united the Albanian princes in the fight against the Ottoman Empire.  Skanderbeg was buried in the cathedral of Lezhë which was dedicated to Saint Nicholas and later used as Selimie Mosque.

Modern era.

Today Lezhë is a growing city. Its proximity to the port of Shëngjin as well as its location on the national road between the Montenegrin border to the North and Tirana to the South makes it an attractive location for industry and business.  Majority of the people from Lezhë descend from the Zadrima, Mirdita and Malësia/Malësi e madhe regions of northwestern Albania. The people from Zadrima and Mirdita are native to Lezhë and the surrounding area. Whilst the Malësor clans from Malësia, such as Kelmendi, Shkreli, Kastrati etc., had settled Lezhë and surrounding areas around 100–300 years ago. In terms of religion, Lezha has a Catholic majority(70-80%) and a Muslim minority(30-20%)

Transportation.

Bus.

There are urban buses throughout the city and international and national buses.

Trains.

Lezhe has a train station not far from the center. The line starts in Durrës and ends in Shkodër. It is functonally but not frequently.

Highways.

The main highway in Lezhe is E762. The E762 stars in Fushë-Krujë and ends in Shkodër. The Durrës-Kukës Highway intersects with E762 in Milot. Also the SH32 intersects in to Lezhe coming from Shëngjin.

Sport.

The association football club is KS Besëlidhja Lezhë. Although primarily concerned with football, KS Besëlidhja also participates in sports such as wrestling and beach volleyball.

Archaeological excavations.

From 2004 an excavation started around the ancient Acropolis of Lissos and the Skanderbeg Memorial, which revealed Hellenistic, Roman and Early Byzantine buildings, tombs and other findings.

Lezhë ( or "Lezhë", ) is a town and municipality in northwest Albania, in the county with the same name.

Lezhë je město ležící ve středu Albánie, 60 km severně od metropole Tirany. Žije zde obyvatel. Městem Lezhë prochází důležitá silnice Tirana–Skadar, hlavní železniční trať. Protéká jím řeka Drim, samo se nachází zhruba 9 km od ústí řeky do moře. Město je centrem okresu Lezhë a kraje Lezhë.


History.

Design and construction.

In the time of Frederick William (1688), shortly after the Thirty Years' War and a century before the gate was constructed, Berlin was a small walled city within a star fort with several named gates: Spandauer Tor, St. Georgen Tor, Stralower Tor, Cöpenicker Tor, Neues Tor, and Leipziger Tor "(see map)". Relative peace, a policy of religious tolerance, and status as capital of the Kingdom of Prussia facilitated the growth of the city. The Brandenburg Gate was not part of the old Berlin Fortress, but one of eighteen gates within the Berlin Customs Wall (), erected in the 1730s, including the old fortified city and many of its then suburbs.  The new gate was commissioned by Frederick William II of Prussia to represent peace and was originally named the Peace Gate (German: "Friedenstor"). It was designed by Carl Gotthard Langhans, the Court Superintendent of Buildings, and built between 1788 and 1791, replacing the earlier simple guardhouses which flanked the original gate in the Customs Wall. The gate consists of twelve Doric columns, six to each side, forming five passageways. Citizens were originally allowed to use only the outermost two on each side. Its design is based on the Propylaea, the gateway to the Acropolis in Athens, Greece, and is consistent with Berlin's history of architectural classicism (first, Baroque, and then neo-Palladian). The gate was the first element of a "new Athens on the River Spree" by architect Langhans. Atop the gate is a Quadriga, a chariot drawn by four horses driven by Victoria, the Roman goddess of victory, sculpted by Johann Gottfried Schadow.

19th and early 20th centuries.

The Brandenburg Gate has played different political roles in German history. After the 1806 Prussian defeat at the Battle of Jena-Auerstedt, Napoleon was the first to use the Brandenburg Gate for a triumphal procession, and took its Quadriga to Paris.  After Napoleon's defeat in 1814 and the Prussian occupation of Paris by General Ernst von Pfuel, the Quadriga was restored to Berlin. It was now redesigned by Karl Friedrich Schinkel for the new role of the Brandenburg Gate as a Prussian triumphal arch. The goddess, now definitely Victoria, was equipped with the Prussian eagle and Iron Cross on her lance with a wreath of oak leaves. The Quadriga faces east, as it did when it was originally installed in 1793. Only the royal family was allowed to pass through the central archway, as well as members of the Pfuel family, from 1814 to 1919. The Kaiser granted this honour to the family in gratitude to Ernst von Pfuel, who had overseen the return of the Quadriga to the top of the gate. In addition, the central archway was also used by the coaches of ambassadors on the single occasion of their presenting their letters of credence to council.  When the Nazis ascended to power, they used the gate as a party symbol. The gate survived World War II and was one of the damaged structures still standing in the Pariser Platz ruins in 1945 (another being the Academy of Fine Arts). The gate was badly damaged with holes in the columns from bullets and nearby explosions. One horse's head from the original quadriga survived, and is today kept in the collection of the Märkisches Museum.

Cold War.

After Germany's surrender and the end of the war, the governments of East Berlin and West Berlin restored it in a joint effort. The holes were patched, but were visible for many years. The gate was located in the Soviet occupation zone, directly next to the border to the British occupation zone, which later became the border between East and West Berlin.  Vehicles and pedestrians could travel freely through the gate until the day after construction began on the Berlin Wall on Barbed Wire Sunday, 13 August 1961. West Berliners gathered on the western side of the gate to demonstrate against the Berlin Wall, among them West Berlin's mayor, Willy Brandt, who had returned from a federal election campaign tour in West Germany earlier the same day. The wall passed directly by the western side of the gate, which was closed throughout the Berlin Wall period, which ended on 22 December 1989.

Post-1989.

When the Revolutions of 1989 occurred and the wall was demolished, the gate symbolized freedom and the desire to unify the city of Berlin. Thousands of people gathered at the wall to celebrate its fall on 9 November 1989. On 22 December 1989, the Brandenburg Gate border crossing was reopened when Helmut Kohl, the West German chancellor, walked through to be greeted by Hans Modrow, the East German prime minister. Demolition of the rest of the wall around the area took place the following year.  During 1990, the quadriga was removed from the gate as part of renovation work carried out by the East German authorities following the fall of the wall in November 1989. Germany was officially reunified in October 1990.  The Brandenburg Gate was privately refurbished on 21 December 2000, at a cost of six million euros. It was once again opened on 3 October 2002 following extensive refurbishment, for the 12th anniversary of German reunification.  Brandenburg Gate became the main venue for the 20th-anniversary celebrations of the fall of the Berlin Wall or "Festival of Freedom" on the evening of 9 November 2009. The high point of the celebrations was when over 1000 colorfully designed foam domino tiles, each over tall, were lined up along the route of the former wall through the city centre. The domino "wall" was then toppled in stages converging here.  The Brandenburg Gate is now again closed to vehicle traffic, and much of Pariser Platz has been turned into a cobblestone pedestrian zone. The gate, along with the broad "Straße des 17. Juni" avenue to the west, is also one of the large public areas in Berlin where over a million people can gather to watch stage shows or party together, watch major sport events shown on huge screens, or see fireworks at midnight on New Year's Eve. After winning the 2014 FIFA World Cup, the German national football team held their victory rally in front of the gate.  It has also hosted street events at 2009 IAAF World Championships in Athletics and will repeat its role in 2018 European Athletics Championships. It is also the usual finish line of the Berlin Marathon.

Political history.

A Soviet flag flew from a flagpole atop the gate from 1945 until 1957, when it was replaced by an East German flag. Since the reunification of Germany in 1990, the flag and the pole have been removed. During the 1953 riots in East Berlin the Soviet flag was torn off by West Germans.  In 1963, U.S. President John F. Kennedy visited the Brandenburg Gate. The Soviets hung large red banners across it to prevent him looking into East Berlin.  In the 1980s, decrying the existence of two German states and two Berlins, West Berlin mayor Richard von Weizsäcker said: "The German question is open as long as the Brandenburg Gate is closed."  On 12 June 1987, U.S. President Ronald Reagan spoke to the West Berlin populace at the Brandenburg Gate, demanding the razing of the Berlin Wall. Addressing the General Secretary of the Communist Party of the Soviet Union, Mikhail Gorbachev, Reagan said,  On 25 December 1989, less than two months after the Berlin Wall began to come down, the conductor Leonard Bernstein conducted the Berlin Philharmonic in a version of the Ninth Symphony of Beethoven at the then newly opened Brandenburg Gate. In the concluding choral movement of the symphony, the "Ode to Joy", the word "Freude" ("Joy") was replaced with "Freiheit" ("Freedom") to celebrate the fall of the Wall and the imminent reunification of Germany.  On 2–3 October 1990, the Gate was the scene of the official ceremony to mark the reunification of Germany. At the stroke of midnight on 3 October, the black-red-gold flag of West Germany—now the flag of a reunified Germany—was raised over the Gate.  On 12 July 1994, U.S. President Bill Clinton spoke at the Gate about peace in post–Cold War Europe.  On 9 November 2009, Chancellor Angela Merkel, walked through Brandenburg Gate with Russia's Mikhail Gorbachev and Poland's Lech Wałęsa as part of the 20-year celebration of tearing down the Berlin Wall.  On 13 August 2011, Germany marked the 50th anniversary of the day the Berlin Wall started to go up with a memorial service and a minute of silence in memory of those who died trying to flee to the West. "It is our shared responsibility to keep the memory alive and to pass it on to the coming generations as a reminder to stand up for freedom and democracy to ensure that such injustice may never happen again," Berlin Mayor Klaus Wowereit said. German Chancellor Angela Merkel—who grew up behind the wall in Germany's communist eastern part—also attended the commemoration. German President Christian Wulff added, "It has been shown once again: Freedom is invincible at the end. No wall can permanently withstand the desire for freedom."  On 19 June 2013, U.S. President Barack Obama spoke at the Gate about nuclear arms reduction and the recently revealed U.S. internet surveillance activities.  On the night of 5 January 2015, the lights illuminating the gate were completely shut off in protest against a protest held by far-right anti-Islamic group Pegida.  In April 2017, Die Zeit noted that the gate was not illuminated in Russian colours after the 2017 Saint Petersburg Metro bombing. The gate was previously illuminated after attacks in Jerusalem and Orlando. The Berlin Senate only lets the gate be illuminated for events in partner cities and cities with a special connection to Berlin.

The Brandenburg Gate (; ) is an 18th-century neoclassical monument in Berlin, built on the orders of Prussian king Frederick William II after the temporary restoration of order during the Batavian Revolution. One of the best-known landmarks of Germany, it was built on the site of a former city gate that marked the start of the road from Berlin to the town of Brandenburg an der Havel, which used to be capital of the Margraviate of Brandenburg.

Braniborská brána, Brandenburger Tor, patří k nejznámějším pamětihodnostem Berlína a k jeho významným symbolům. Braniborská brána stojí v centru města na rozhraní dvou známých a velkých tříd: na západ od ní počíná "Straße des 17. Juni" (Ulice 17. června), na východ od brány pak pokračuje historická třída "Unter den Linden" (Pod lípami). S touto stavební památkou je spojeno mnoho historických událostí města.


Etymology.

Saint Roch is given different names in various languages:

Traditional biography.

According to his "Acta" and his "vita" in the "Golden Legend," he was born at Montpellier, at that time "upon the border of France," as the "Golden Legend" has it, the son of the noble governor of that city. Even his birth was accounted a miracle, for his noble mother had been barren until she prayed to the Virgin Mary. Miraculously marked from birth with a red cross on his breast that grew as he did, he early began to manifest strict asceticism and great devoutness; on days when his "devout mother fasted twice in the week, and the blessed child Rocke abstained him twice also, when his mother fasted in the week, and would suck his mother but once that day."  On the death of his parents in his twentieth year he distributed all his worldly goods among the poor like Francis of Assisi—though his father on his deathbed had ordained him governor of Montpellier—and set out as a mendicant pilgrim for Rome. Coming into Italy during an epidemic of plague, he was very diligent in tending the sick in the public hospitals at Acquapendente, Cesena, Rimini, Novara, and Rome, and is said to have effected many miraculous cures by prayer and the sign of the cross and the touch of his hand. At Rome, according to the "Golden Legend" he preserved the "cardinal of Angleria in Lombardy" by making the mark of the cross on his forehead, which miraculously remained. Ministering at Piacenza he himself finally fell ill. He was expelled from the town; and withdrew into the forest, where he made himself a hut of boughs and leaves, which was miraculously supplied with water by a spring that arose in the place; he would have perished had not a dog belonging to a nobleman named Gothard Palastrelli supplied him with bread and licked his wounds, healing them. Count Gothard, following his hunting dog that carried the bread, discovered Saint Roch and became his acolyte.  On his return incognito to Montpellier he was arrested as a spy (by orders of his own uncle) and thrown into prison, where he languished five years and died on 16 August 1327, without revealing his name, to avoid worldly glory. (Evidence suggests, as mentioned earlier, that the previous events occurred, instead at Voghera in the 1370s.) After his death, according to the "Golden Legend";  The townspeople recognized him as well by his birthmark; he was soon canonized in the popular mind, and a great church erected in veneration.  The date (1327) asserted by Francesco Diedo for Saint Roch's death would precede the traumatic advent of the Black Death in Europe (1347–49) after long centuries of absence, for which a rich iconography of the plague, its victims and its protective saints was soon developed, in which the iconography of Roche finds its historical place: previously the "topos" did not exist. In contrast, however, St. Roch of Montpellier cannot be dismissed based on dates of a specific plague event. In medieval times, the term "plague" was used to indicate a whole array of illnesses and epidemics.  The first literary account is an undated "Acta" that is labeled, by comparison with the longer, elaborated accounts that were to follow, "Acta Breviora", which relies almost entirely on standardized hagiographic "topoi" to celebrate and promote the cult of Roch.  The story that when the Council of Constance was threatened with plague in 1414, public processions and prayers for the intercession of Roch were ordered, and the outbreak ceased, is provided by Francesco Diedo, the Venetian governor of Brescia, in his "Vita Sancti Rochi", 1478. The cult of Roch gained momentum during the bubonic plague that passed through northern Italy in 1477–79.

Veneration.

His popularity, originally in central and northern Italy and at Montpellier, spread through Spain, France, Lebanon, the Low Countries, Brazil and Germany, where he was often interpolated into the roster of the Fourteen Holy Helpers, whose veneration spread in the wake of the Black Death. The 16th-century Scuola Grande di San Rocco and the adjacent church of San Rocco were dedicated to him by a confraternity at Venice, where his body was said to have been surreptitiously translated and was triumphantly inaugurated in 1485; the "Scuola Grande" is famous for its sequence of paintings by Tintoretto, who painted St. Roch in glory in a ceiling canvas (1564).  We know for certain that the body of St. Roch was carried from Voghera, instead of Montpellier as previously thought, to Venice in 1485. Pope Alexander VI (1492–1503) built a church and a hospital in his honor. Pope Paul III (1534–1549) instituted a confraternity of St. Roch. This was raised to an arch-confraternity in 1556 by Pope Paul IV; it still thrives today. Saint Roch had not been officially recognized as yet, however. In 1590 the Venetian ambassador at Rome reported back to the Serenissima that he had been repeatedly urged to present the witnesses and documentation of the life and miracles of San Rocco, already deeply entrenched in the Venetian life, because Pope Sixtus V "is strong in his opinion either to canonize him or else to remove him from the ranks of the saints;" the ambassador had warned a cardinal of the general scandal that would result if the widely venerated San Rocco were impugned as an impostor. Sixtus did not pursue the matter but left it to later popes to proceed with the canonization process. His successor, Pope Gregory XIV (1590–1591), added Roch of Montpellier, who had already been memorialized in the Holy Sacrifice of the Mass for two centuries, to the Roman Catholic Church Martyrology, thereby fixing August 16 as his universal feast day.  Numerous brotherhoods have been instituted in his honor. He is usually represented in the garb of a pilgrim, often lifting his tunic to demonstrate the plague sore, or bubo, in his thigh, and accompanied by a dog carrying a loaf in its mouth. The Third Order of Saint Francis, by tradition, claims him as a member and includes his feast on its own calendar of saints, observing it on August 17.

Saint Roch in art.

Following the Black Death, especially the Italian plague epidemic of 1477–79, new images of Christian martyrs and saints appeared and Saint Roch gained new fame and popularity. The religious art of the time emphasized the importance of the saint to plague-ridden Christians.  The new plague-related images of Saint Roch were drawn from a variety of sources. Plague texts dating from ancient and classical times, as well as Christian, scientific and folk beliefs, all contributed to this emerging visual tradition. Some of the most popular symbols of plague were swords, darts, and most especially arrows. There was also a prevalence of memento mori themes, dark clouds, and astrological signs (signa magna) such as comets, which were often referenced by physicians and writers of plague tracts as causes of plague. The physical symptoms of plague – a raised arm, a tilted head, or a collapsed body – began to symbolize plague in post-Black Death painting.  Plague saints offered hope and healing before, during, and after times of plague. A specific style of painting, the plague votive, was considered a talisman for warding off plague. It portrayed a particular saint as an intercessor between God and the person or persons who commissioned the painting – usually a town, government, lay confraternity, or religious order to atone for the "collective guilt" of the community.  These plague votives worked as a psychological defense against disease in which people attempted to manipulate their situation through requesting the intercession of a saint against the arrows of plague. Rather than a society depressed and resigned to repeated epidemics, these votives represent people taking positive steps to regain control over their environment. Paintings of St. Roch represent the confidence in which renaissance worshipers sought to access supernatural aid in overcoming the ravages of plague.  The very abundance of means by which people invoked the aid of the celestial court is essential in understanding Renaissance responses to the disease. Rather than depression or resignation, people "possessed a confidence that put even an apocalyptic disaster of the magnitude of the Black Death into perspective of God's secure and benevolent plan for humankind."  The plague votives functioned both to request intercessory aid from plague saints and to provide catharsis for a population that had just witnessed the profound bodily destruction of the plague. By showing plague saints such as St. Roch and St. Sebastian, votives influenced the distribution of God's mercy by invoking the memory of the human suffering experienced by Christ during the Passion. In the art of St. Roch after 1477 the saint displayed the wounds of his martyrdom without evidence of pain or suffering. Roch actively lifted his clothing to display the plague bubo in his thigh. This display of his plague bubo showed that "he welcomed his disease as a divinely sent opportunity to imitate the sufferings of Christ... [his] patient endurance [of the physical suffering of plague was] a form of martyrdom."  Roch's status as a pilgrim who suffered plague is paramount in his iconography. "The sight of Roch scarred by the plague yet alive and healthy must have been an emotionally-charged image of a promised cure. Here was literal proof that one could survive the plague, a saint who had triumphed over the disease in his own flesh."

Saint Roch in literature.

F. T. Prince published a long monologue from the perspective of Saint Roch's dog entitled 'His Dog and Pilgrim' in his 1983 collection "Later On".

Saint Roch or Rocco (lived c. 1348 – 15/16 August 1376/79 (traditionally c. 1295 – 16 August 1327) was a Catholic saint, a confessor whose death is commemorated on 16 August and 9 September in Italy; he is specially invoked against the plague. He may also be called Rock in English, and has the designation of St Rollox in Glasgow, Scotland, said to be a corruption of St Roch's Loch, which referred to a small loch once near a chapel dedicated to St. Roch in 1506.

Svatý Roch z Montpellieru (okolo 1295 Montpellier – 16. srpna 1327 Montpellier) je francouzský katolický světec, uctívaný jako patron proti moru.


Biblical narrative.

In the Hebrew Bible, the Kingdom of Israel has been referred to as the "House of Joseph". It is also frequently referenced (particularly in poetry) as Ephraim, the tribe whose territory housed the capital cities and the royal families. It has also been referred to as "Israel in Samaria".  According to the Hebrew Bible, the territory of the Kingdom of Israel comprised the territories of the tribes of Zebulun, Issachar, Asher, Naphtali, Dan, Manasseh, Ephraim, Reuben and Gad. Its capital was Samaria according to the Book of Isaiah.

Biblical background.

United Monarchy.

The United Kingdom of Israel and Judah is said to have existed from about 1030 to about 930 BCE. It was a union of all the twelve Israelite tribes living in the area that presently approximates modern Israel and the other Levantine territories including much of western Jordan, and western Syria.

Division.

After the death of Solomon in about 931 BCE, most of the Israelite tribes (ten Northern tribes) except for Judah and Benjamin refused to accept Rehoboam, the son and successor of Solomon, as their king. The rebellion against Rehoboam arose after he refused to lighten the burden of taxation and services that his father had imposed on his subjects.  Jeroboam, who was not of the Davidic line, was sent forth from Egypt by the malcontents. The Tribe of Ephraim and all Israel raised the old cry, "Every man to his tents, O Israel". Rehoboam fled to Jerusalem, and in 930 BCE (some date it in 920 BCE), Jeroboam was proclaimed king over all Israel at Shechem. After the revolt at Shechem at first only the tribe of Judah remained loyal to the house of David. But very soon after the tribe of Benjamin joined Judah. The northern kingdom continued to be called the Kingdom of Israel or Israel, while the southern kingdom was called the Kingdom of Judah. also says that members of the tribes of Ephraim, Manasseh and Simeon fled to Judah during the reign of Asa of Judah.

History.

Early kings and Omride dynasty.

Shechem was the first capital of the Kingdom of Israel. Afterwards it was Tirzah. King Omri built his capital in Samaria (1 Kings 16:24), which continued as such until the destruction of the Kingdom by the Assyrians ().  Today, among archaeologists, Samaria is one of the most universally accepted archaeological sites from the biblical period At around 850 BCE, the Mesha Stele, written in Old Hebrew alphabet, records a victory of King Mesha of Moab against king Omri of Israel and his son Ahab.

Relations between the kingdoms of Israel and Judah.

According to the Bible, for the first sixty years, the kings of Judah tried to re-establish their authority over the northern kingdom, and there was perpetual war between them. For the following eighty years, there was no open war between them, and, for the most part, they were in friendly alliance, co-operating against their common enemies, especially against Damascus.  The conflict between Israel and Judah was resolved when Jehoshaphat, King of Judah, allied himself with the house of Ahab through marriage. Later, Jehosophat's son and successor, Jehoram of Judah, married Ahab's daughter Athaliah, cementing the alliance. However, the sons of Ahab were slaughtered by Jehu following his coup d'état around 840 BCE.

Destruction of the kingdom.

In c. 732 BCE, Pekah of Israel, while allied with Rezin, king of Aram, threatened Jerusalem. Ahaz, king of Judah, appealed to Tiglath-Pileser III, the king of Assyria, for help. After Ahaz paid tribute to Tiglath-Pileser Tiglath-Pileser sacked Damascus and Israel, annexing Aram and territory of the tribes of Reuben, Gad and Manasseh in Gilead including the desert outposts of Jetur, Naphish and Nodab. People from these tribes including the Reubenite leader, were taken captive and resettled in the region of the Khabur River system. Tiglath-Pilesar also captured the territory of Naphtali and the city of Janoah in Ephraim and an Assyrian governor was placed over the region of Naphtali. According to and, the population of Aram and the annexed part of Israel was deported to Assyria. The remainder of the northern kingdom of Israel continued to exist within the reduced territory as an independent kingdom until around 720 BCE, when it was again invaded by Assyria and the rest of the population deported. During the three-year siege of Samaria in the territory of Ephraim by the Assyrians, Shalmaneser V died and was succeeded by Sargon II, who himself records the capture of that city thus: "Samaria I looked at, I captured; 27,280 men who dwelt in it I carried away" into Assyria. Thus, around 720 BCE, after two centuries, the kingdom of the ten tribes came to an end. Some of the Israelite captives were resettled in the Khabur region, and the rest in the land of the Medes, thus establishing Hebrew communities in Ecbatana and Rages. The Book of Tobit additionally records that Sargon had taken other captives from the northern kingdom to the Assyrian capital of Nineveh, in particular Tobit from the town of Thisbe in Naphtali.  The Hebrew Bible relates that the population of the Kingdom of Israel was exiled, becoming known as the Ten Lost Tribes. To the south, the Tribe of Judah, the Tribe of Simeon (that was "absorbed" into Judah), the Tribe of Benjamin and the people of the Tribe of Levi, who lived among them of the original Israelite nation, remained in the southern Kingdom of Judah. The Kingdom of Judah continued to exist as an independent state until 586 BCE, when it was conquered by the Neo-Babylonian Empire.

Samaritan version.

The Samaritan version to the events claims that actually much of the population of the Northern Kingdom of Israel remained in place upon the Exile, including the Tribes of Naphtali, Menasseh, Benjamin and Levi - being the progenitors of the Samaritans. In their book "The Bible Unearthed", authors Israel Finkelstein and Neil Asher Silberman estimate that only a fifth of the population (about 40,000) were actually resettled out of the area during the two deportation periods under Tiglath-Pileser III and Sargon II. Many of the Northern Tribes also fled south to Jerusalem, which appears to have expanded in size five-fold during this period, requiring a new wall to be built, and a new source of water Siloam to be provided by King Hezekiah.

Medieval Rabbinic fable.

In medieval Rabbinic fable, the concept of the ten tribes who were taken away from the House of David (who continued the rule of the southern kingdom of Judah), becomes confounded with accounts of the Assyrian deportations leading to the myth of the "Ten Lost Tribes".

Recorded history.

The recorded history differs from the Rabbinic fable: No record exists of the Assyrians having exiled people from Dan, Asher, Issachar, Zebulun or western Manasseh. Descriptions of the deportation of people from Reuben, Gad, Manasseh in Gilead, Ephraim and Naphtali indicate that only a portion of these tribes were deported and the places to which they were deported are known locations given in the accounts. The deported communities are mentioned as still existing at the time of the composition of the Books of Kings and Chronicles and did not disappear by assimilation. explicitly mentions northern Israelites who had been spared by the Assyrians, in particular people of Ephraim, Manasseh, Asher, Issachar and Zebulun, and how members of the latter three returned to worship at the Temple in Jerusalem during the reign of Hezekiah.

Religion.

The religious climate of the Kingdom of Israel appears to have followed two major trends. The first, that of worship of Yahweh, and the second that of worship of Baal as detailed in the Hebrew Bible () and in the Baal cycle discovered at Ugarit.  According to the Hebrew Bible Jeroboam built two places of worship, one at Bethel and one at far northern Dan, as alternatives to the Temple in Jerusalem.() He did not want the people of his kingdom to have religious ties to Jerusalem, the capital city of the rival Kingdom of Judah. He erected golden bulls at the entrance to the Temples to represent the national god. The Hebrew Bible, written from the perspective of scribes in Jerusalem, referred to these acts as the way of Jeroboam or the errors of Jeroboam. ()  The Bible states that Ahab allowed the cult worship of Baal to become an acceptable religion of the kingdom. His wife Jezebel was a devotee to Baal worship. ()

List of proposed Assyrian references to Kingdom of Israel (Samaria).

The table below lists all the historical references to the Kingdom of Israel (Samaria) in Assyrian records. King Omri's name takes the Assyrian shape of "Humri", his kingdom or dynasty that of Bit Humri or alike - the "House of Humri/Omri".

According to the Hebrew Bible, the Kingdom of Israel (), was one of two successor states to the former United Kingdom of Israel and Judah. Historians often refer to the Kingdom of Israel as the "Northern Kingdom" or as the "Kingdom of Samaria" to differentiate it from the Southern Kingdom of Judah.

Izraelské království (někdy také „severní království“ oproti „jižnímu“ judskému) je státní útvar, který existoval mezi lety 930–722 př. n. l. převážně v oblasti Samařska (severní území dnešního Izraele) s hlavním městem Samaří.


History.

The term's origins are in ecclesiastical language from the Middle Ages: when a non-Italian was elected to the papacy, he was said to be "papa ultramontano", that is, a pope from beyond the mountains (the Alps). Foreign students at medieval Italian universities were also referred to as "ultramontani".  The word was revived but the meaning reversed after the Protestant Reformation in France, to indicate the "man beyond the mountains" located in Italy. In France, the name "ultramontain" was applied to people who supported papal authority in French affairs, as opposed to the Gallican and Jansenist factions of the indigenous French Catholic Church. The term was intended to be insulting, or at least to imply a lack of true patriotism. From the 17th century, ultramontanism became closely associated with the Jesuits.  In the 18th century the word passed to Austria (Josephinism and Febronianism), where it acquired a much wider significance, being applicable to all the conflicts between church and state, the supporters of the Church being called "ultramontanes". In Great Britain and Ireland, it was a reaction to Cisalpinism, the stance of moderate lay Catholics who sought to make patriotic concessions to the Protestant state to achieve Catholic emancipation.  In Canada, the majority of Catholic clergy despised the French Revolution and its anti-clerical bias and looked to Rome for both spiritual and political guidance. There were many laymen and laywomen who supported these ideals as key to preserving Canadian institutions and values. For this reason they were called ultramontanists. The ultramontanes distrusted both the Protestant anglophone and francophone politicians, but the Church found it easier to deal with British governors, who appreciated the role of the Church in containing dissent, than with the francophone liberal professionals who were secularists.

First Vatican Council.

According to Catholic academic Jeffrey P. von Arx,The threat to the Catholic Church and the papacy through the 19th century was real, and the church’s reaction to that threat was understandable. Indeed, the church remained threatened on all sides. On the left, secular liberals sought to reduce or eliminate the role of the church in public life and civil society (by suppressing church schools, for example, and expelling religious congregations). The more radical heirs of the revolution and the socialists and communists into whom they evolved remained committed to the church’s utter destruction. But the threat was also from the nationalist right. Otto von Bismarck’s Kulturkampf was aimed directly at the Catholic Church, imposing state supervision of Catholic schools and seminaries and government appointment of bishops with no reference to Rome.  The response was a condemnation of Gallicanism as heretical, [W]e condemn and reject the opinions of those who hold that this communication of the supreme head with pastors and flocks may be lawfully obstructed; or that it should be dependent on the civil power, which leads them to maintain that what is determined by the apostolic see or by its authority concerning the government of the church, has no force or effect unless it is confirmed by the agreement of the civil authority. The Council also asserted papal primacy. In July 1870, it issued the Dogmatic constitution "Pastor aeternus", defining four doctrines of the Catholic faith: the apostolic primacy conferred on Peter, the perpetuity of this primacy in the Roman pontiffs, the meaning and power of the papal primacy, and Papal infallibility. [W]e teach and declare that, by divine ordinance, the Roman Church possesses a pre-eminence of ordinary power over every other Church, and that this jurisdictional power of the Roman Pontiff is both episcopal and immediate. Both clergy and faithful, of whatever rite and dignity, both singly and collectively, are bound to submit to this power by the duty of hierarchical subordination and true obedience, and this not only in matters concerning faith and morals, but also in those which regard the discipline and government of the Church throughout the world."  Von Arx compares this to "...the great empires and national states of the 19th century, which used new means of communication and transportation to consolidate power, enforce unity and build bureaucracies." "Cardinal Henry Edward Manning in Great Britain thought unity and discipline within the church were of the utmost importance in protecting the church and advancing its interests in a liberal, democratic state, and so he was one of the strongest advocates of the ultramontane position." The English bishops at the Council were characterized by their ultramontanism and described as "being more Catholic than the Pope himself".

Reaction.

Other Christian groups outside the Catholic Church declared this as the triumph of what they termed "the heresy of ultramontanism". It was specifically decried in the "Declaration of the Catholic Congress at Munich", in the Theses of Bonn, and in the Declaration of Utrecht, which became the foundational documents of Old Catholics ("Altkatholische") who split with Rome over the declaration on infallibility and supremacy, joining the Old Episcopal Order Catholic See of Utrecht, which had been independent from Rome since 1723.  As with previous pronouncements by the pope, liberals across Europe were outraged by the doctrine of infallibility and many countries reacted with laws to counter the influence of the church. The term "ultramontanism" was revived during the French Third Republic (1870–1940) as a pejorative way to describe policies that went against "laïcité" a concept rooted in the French Revolution. French philosopher Jacques Maritain, noted the distinction between the models found in France and the separation of church and state in the United States in the mid-twentieth century. He considered the US model of that time to be more amicable because it had both "sharp distinction and actual cooperation" between church and state, what he called "an historical treasure" and admonished the United States, "Please to God that you keep it carefully, and do not let your concept of separation veer round to the European one."  After Italian Unification and the abrupt (and unofficial) end of the First Vatican Council in 1870 because of the outbreak of the Franco-Prussian War, the ultramontanist movement and the opposing conciliarism became obsolete to a large extent. However, some very extreme tendencies of a minority of adherents to ultramontanism – especially those attributing to the Roman pontiff, even in his private opinions, absolute infallibility even in matters beyond faith and morals, and impeccability – survived and were eagerly used by opponents of the Catholic Church and papacy before the Second Vatican Council (1962–1965) for use in their propaganda. These extreme tendencies, however, were never supported by the First Vatican Council's dogma of 1870 of papal infallibility and primacy, but were rather inspired by erroneous private opinions of some Roman Catholic laymen who tend to identify themselves completely with the Holy See.  At the Second Vatican Council's Dogmatic Constitution on the Church "Lumen gentium", the Roman Catholic Church's teaching on the authority of the pope, bishops and councils was further elaborated. The post-conciliar position of the Apostolic See did not deny any of the previous doctrines of papal infallibility or papal primacy; rather, it shifted emphasis from structural and organizational authority to doctrinal teaching authority (also known as the "magisterium"). Papal "magisterium", i.e. papal teaching authority, was defined in "lumen gentium" No. 25 and later codified in the 1983 revision of Canon Law.

Controversy.

Some, such as the former Cardinal Joseph Ratzinger, claim the Catholic Social Teaching (see Distributism) of subsidiarity can overrun ultramontanism and has the potential to decentralize the Roman Catholic Church, whereas others defend it as merely a bureaucratic adjustment to give more pastoral responsibility to local bishops and priests of local parishes.  Challenges to ultramontanism have remained strong within and outside Roman jurisdiction. Ultramontanism has particularly overshadowed ecumenical work between the Roman Catholic Church and both Lutherans and Anglicans. The joint Anglican-Roman Catholic International Consultation published "The Gift of Authority" in 1999, highlights agreements and differences on these issues.

Position of other traditional churches.

Ultramontanism is distinct from the positions adopted by the other traditional churches, particularly the Anglican communion, Eastern Orthodox communion, the Oriental Orthodox communion, or the Church of the East. These churches regard the pope as having been "primus inter pares" when the churches were united in full communion, and generally still acknowledge that status today, albeit in an impaired form due to disunity; similarly they do not recognize the doctrines of infallibility or the pope's alleged universal jurisdiction over patriarchates and autocephalous churches other than that of Rome itself, except insofar as this is part of the concept of "primus inter pares".  In the joint agreed statement "The Gift of Authority" (1999) The Anglican communion and the Roman Catholic church were agreed on the collegial nature of the life and work of bishops. Similarly both churches acknowledged the role of episcopal primacy within the college of bishops. On the question of the universal primacy of the Pope, the joint report found common ground, and stated that a "particular conclusion" of their discussions had been "that Anglicans be open to and desire a recovery and re-reception under certain clear conditions of the exercise of universal primacy by the Bishop of Rome"; nonetheless a clear distinction remained between the Anglican view of a universal primacy exercised within a universal collegiality, and the Roman Catholic view of a universal primacy with actual universal jurisdiction.

Ultramontanism is a clerical political conception within the Catholic Church that places strong emphasis on the prerogatives and powers of the Pope.

Pojem ultramontanismus (lat. "ultra montes", za horami, myšleno Alpami) označuje politiku podpory těch „za horami“ (Alpami), tj. papeže v domácí politice evropských států, zvláště v 19. století. V uzším slova smyslu znamená ultramontanismus souhlas s nadřazeností papežské autority nad místními autoritami jak světskými, tak duchovními (tj. diecézními biskupy).


Definition.

A Banach space is a vector space over any scalar field K that is equipped with a norm formula_1 and that is complete with respect to the distance function induced by the norm, that is to say, for every Cauchy sequence in, there exists an element in such that  or equivalently:  The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space is a Banach space if and only if each absolutely convergent series in converges in,  Completeness of a normed space is preserved if the given norm is replaced by an equivalent one.  All norms on a finite-dimensional vector space are equivalent. Every finite-dimensional normed space over or is a Banach space.

General theory.

Linear operators, isomorphisms.

If and are normed spaces over the same ground field, the set of all continuous -linear maps is denoted by. In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space to another normed space is continuous if and only if it is bounded on the closed unit ball of. Thus, the vector space can be given the operator norm  For a Banach space, the space is a Banach space with respect to this norm.  If is a Banach space, the space forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.  If and are normed spaces, they are isomorphic normed spaces if there exists a linear bijection such that and its inverse are continuous. If one of the two spaces or is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces and are isometrically isomorphic if in addition, is an isometry, i.e., for every in. The Banach–Mazur distance between two isomorphic but not isometric spaces and gives a measure of how much the two spaces and differ.

Basic notions.

Every normed space can be isometrically embedded in a Banach space. More precisely, for every normed space, there exist a Banach space and a mapping such that T is an isometric mapping and is dense in. If is another Banach space such that there is an isometric isomorphism from onto a dense subset of, then is isometrically isomorphic to.  This Banach space is the completion of the normed space. The underlying metric space for is the same as the metric completion of, with the vector space operations extended from to. The completion of is often denoted by formula_6.  The cartesian product of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as  and give rise to isomorphic normed spaces. In this sense, the product (or the direct sum ) is complete if and only if the two factors are complete.  If is a closed linear subspace of a normed space, there is a natural norm on the quotient space,  The quotient is a Banach space when is complete. The quotient map from onto, sending in to its class, is linear, onto and has norm, except when, in which case the quotient is the null space.  The closed linear subspace of is said to be a complemented subspace of if is the range of a bounded linear projection from onto. In this case, the space is isomorphic to the direct sum of and, the kernel of the projection.  Suppose that and are Banach spaces and that. There exists a canonical factorization of as  where the first map is the quotient map, and the second map sends every class in the quotient to the image in. This is well defined because all elements in the same class have the same image. The mapping is a linear bijection from onto the range, whose inverse need not be bounded.

Classical spaces.

Basic examples of Banach spaces include: the spaces and their special cases, the sequence spaces that consist of scalar sequences indexed by ; among them, the space of absolutely summable sequences and the space of square summable sequences; the space of sequences tending to zero and the space of bounded sequences; the space of continuous scalar functions on a compact Hausdorff space, equipped with the max norm,  According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some. For every separable Banach space, there is a closed subspace of such that.  Any Hilbert space serves as an example of a Banach space. A Hilbert space on is complete for a norm of the form  where  is the inner product, linear in its first argument that satisfies the following:  For example, the space is a Hilbert space.  The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others.

Banach algebras.

A Banach algebra is a Banach space over or, together with a structure of algebra over, such that the product map "A" × "A" ∋ is continuous. An equivalent norm on can be found so that for all.

Dual space.

If is a normed space and the underlying field (either the real or the complex numbers), the continuous dual space is the space of continuous linear maps from into, or continuous linear functionals. The notation for the continuous dual is in this article. Since is a Banach space (using the absolute value as norm), the dual is a Banach space, for every normed space.  The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.  In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional. An important special case is the following: for every vector in a normed space, there exists a continuous linear functional on such that  When is not equal to the vector, the functional must have norm one, and is called a norming functional for.  The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane. The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.  A subset in a Banach space is total if the linear span of is dense in. The subset is total in if and only if the only continuous linear functional that vanishes on is the functional: this equivalence follows from the Hahn–Banach theorem.  If is the direct sum of two closed linear subspaces and, then the dual of is isomorphic to the direct sum of the duals of and. If is a closed linear subspace in, one can associate the "orthogonal of" in the dual,  The orthogonal is a closed linear subspace of the dual. The dual of is isometrically isomorphic to. The dual of is isometrically isomorphic to.  The dual of a separable Banach space need not be separable, but:  When is separable, the above criterion for totality can be used for proving the existence of a countable total subset in.

Weak topologies.

The weak topology on a Banach space is the coarsest topology on for which all elements in the continuous dual space are continuous. The norm topology is therefore finer than the weak topology. It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed. A norm-continuous linear map between two Banach spaces and is also weakly continuous, i.e., continuous from the weak topology of to that of.  If is infinite-dimensional, there exist linear maps which are not continuous. The space of all linear maps from to the underlying field (this space is called the algebraic dual space, to distinguish it from ) also induces a topology on which is finer than the weak topology, and much less used in functional analysis.  On a dual space, there is a topology weaker than the weak topology of, called weak* topology. It is the coarsest topology on for which all evaluation maps, are continuous. Its importance comes from the Banach–Alaoglu theorem.  The Banach–Alaoglu theorem depends on Tychonoff's theorem about infinite products of compact spaces. When is separable, the unit ball of the dual is a metrizable compact in the weak* topology.

Examples of dual spaces.

The dual of is isometrically isomorphic to : for every bounded linear functional on, there is a unique element such that  The dual of is isometrically isomorphic to. The dual of is isometrically isomorphic to when and.  For every vector in a Hilbert space, the mapping  defines a continuous linear functional on. The Riesz representation theorem states that every continuous linear functional on is of the form for a uniquely defined vector in. The mapping is an antilinear isometric bijection from onto its dual. When the scalars are real, this map is an isometric isomorphism.  When is a compact Hausdorff topological space, the dual of is the space of Radon measures in the sense of Bourbaki. The subset of consisting of non-negative measures of mass 1 (probability measures) is a convex w*-closed subset of the unit ball of. The extreme points of are the Dirac measures on. The set of Dirac measures on, equipped with the w*-topology, is homeomorphic to.  The result has been extended by Amir and Cambern to the case when the multiplicative Banach–Mazur distance between and is. The theorem is no longer true when the distance is.  In the commutative Banach algebra, the maximal ideals are precisely kernels of Dirac mesures on,  More generally, by the Gelfand–Mazur theorem, the maximal ideals of a unital commutative Banach algebra can be identified with its characters—not merely as sets but as topological spaces: the former with the hull-kernel topology and the latter with the w*-topology. In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual.  Not every unital commutative Banach algebra is of the form for some compact Hausdorff space. However, this statement holds if one places in the smaller category of commutative C*-algebras. Gelfand's representation theorem for commutative C*-algebras states that every commutative unital "C"*-algebra is isometrically isomorphic to a space. The Hausdorff compact space here is again the maximal ideal space, also called the spectrum of in the C*-algebra context.

Bidual.

If is a normed space, the (continuous) dual of the dual is called bidual, or second dual of. For every normed space, there is a natural map,  This defines as a continuous linear functional on, i.e., an element of. The map is a linear map from to. As a consequence of the existence of a norming functional for every in, this map is isometric, thus injective.  For example, the dual of is identified with, and the dual of is identified with, the space of bounded scalar sequences. Under these identifications, is the inclusion map from to. It is indeed isometric, but not onto.  If is surjective, then the normed space is called reflexive (see below). Being the dual of a normed space, the bidual is complete, therefore, every reflexive normed space is a Banach space.  Using the isometric embedding, it is customary to consider a normed space as a subset of its bidual. When is a Banach space, it is viewed as a closed linear subspace of. If is not reflexive, the unit ball of is a proper subset of the unit ball of. The Goldstine theorem states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual. In other words, for every in the bidual, there exists a net in so that  The net may be replaced by a weakly*-convergent sequence when the dual is separable. On the other hand, elements of the bidual of that are not in cannot be weak*-limit of "sequences" in, since is weakly sequentially complete.

Banach's theorems.

Here are the main general results about Banach spaces that go back to the time of Banach's book () and are related to the Baire category theorem. According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.  The Banach–Steinhaus theorem is not limited to Banach spaces. It can be extended for example to the case where is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood of in such that all in are uniformly bounded on,  This result is a direct consequence of the preceding "Banach isomorphism theorem" and of the canonical factorization of bounded linear maps.  This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from onto sending to the sum.

Reflexivity.

The normed space is called reflexive when the natural map  is surjective. Reflexive normed spaces are Banach spaces.  This is a consequence of the Hahn–Banach theorem. Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space onto the Banach space, then is reflexive.  Indeed, if the dual of a Banach space is separable, then is separable. If is reflexive and separable, then the dual of is separable, so is separable.  Hilbert spaces are reflexive. The spaces are reflexive when. More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem. The spaces are not reflexive. In these examples of non-reflexive spaces, the bidual is "much larger" than. Namely, under the natural isometric embedding of into given by the Hahn–Banach theorem, the quotient is infinite-dimensional, and even nonseparable. However, Robert C. James has constructed an example of a non-reflexive space, usually called ""the James space"" and denoted by "J", such that the quotient is one-dimensional. Furthermore, this space is isometrically isomorphic to its bidual.  When is reflexive, it follows that all closed and bounded convex subsets of are weakly compact. In a Hilbert space, the weak compactness of the unit ball is very often used in the following way: every bounded sequence in has weakly convergent subsequences.  Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems. For example, every convex continuous function on the unit ball of a reflexive space attains its minimum at some point in.  As a special case of the preceding result, when is a reflexive space over, every continuous linear functional in attains its maximum on the unit ball of. The following theorem of Robert C. James provides a converse statement.  The theorem can be extended to give a characterization of weakly compact convex sets.  On every non-reflexive Banach space, there exist continuous linear functionals that are not "norm-attaining". However, the Bishop–Phelps theorem states that norm-attaining functionals are norm dense in the dual of.

Weak convergences of sequences.

A sequence in a Banach space is weakly convergent to a vector if converges to for every continuous linear functional in the dual. The sequence is a weakly Cauchy sequence if converges to a scalar limit, for every in. A sequence in the dual is weakly* convergent to a functional if converges to for every in. Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.  When the sequence in is a weakly Cauchy sequence, the limit above defines a bounded linear functional on the dual, i.e., an element of the bidual of, and is the limit of in the weak*-topology of the bidual. The Banach space is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in. It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.  An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the vector. The unit vector basis of, or of, is another example of a weakly null sequence, i.e., a sequence that converges weakly to. For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to.  The unit vector basis of is not weakly Cauchy. Weakly Cauchy sequences in are weakly convergent, since -spaces are weakly sequentially complete. Actually, weakly convergent sequences in are norm convergent. This means that satisfies Schur's property.

Results involving the basis.

Weakly Cauchy sequences and the basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.  A complement to this result is due to Odell and Rosenthal (1975).  By the Goldstine theorem, every element of the unit ball of is weak*-limit of a net in the unit ball of. When does not contain, every element of is weak*-limit of a "sequence" in the unit ball of.  When the Banach space is separable, the unit ball of the dual, equipped with the weak*-topology, is a metrizable compact space, and every element in the bidual defines a bounded function on :  This function is continuous for the compact topology of if and only if is actually in, considered as subset of. Assume in addition for the rest of the paragraph that does not contain. By the preceding result of Odell and Rosenthal, the function is the pointwise limit on of a sequence of continuous functions on, it is therefore a first Baire class function on. The unit ball of the bidual is a pointwise compact subset of the first Baire class on.

Sequences, weak and weak* compactness.

When is separable, the unit ball of the dual is weak*-compact by Banach–Alaoglu and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences. This applies to separable reflexive spaces, but more is true in this case, as stated below.  The weak topology of a Banach space is metrizable if and only if is finite-dimensional. If the dual is separable, the weak topology of the unit ball of is metrizable. This applies in particular to separable reflexive Banach spaces. Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.  A Banach space is reflexive if and only if each bounded sequence in has a weakly convergent subsequence.  A weakly compact subset in is norm-compact. Indeed, every sequence in has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of.

Schauder bases.

A Schauder basis in a Banach space is a sequence of vectors in "X" with the property that for every vector in, there exist "uniquely" defined scalars depending on, such that  Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.  It follows from the Banach–Steinhaus theorem that the linear mappings are uniformly bounded by some constant. Let denote the coordinate functionals which assign to every in the coordinate of in the above expansion. They are called biorthogonal functionals. When the basis vectors have norm, the coordinate functionals have norm in the dual of.  Most classical separable spaces have explicit bases. The Haar system is a basis for. The trigonometric system is a basis in when. The Schauder system is a basis in the space. The question of whether the disk algebra has a basis remained open for more than forty years, until Bočkarev showed in 1974 that admits a basis constructed from the Franklin system.  Since every vector in a Banach space with a basis is the limit of, with of finite rank and uniformly bounded, the space satisfies the bounded approximation property. The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.  Robert C. James characterized reflexivity in Banach spaces with a basis: the space with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete. In this case, the biorthogonal functionals form a basis of the dual of.

Tensor product.

Let and be two -vector spaces. The tensor product of and is a -vector space with a bilinear mapping which has the following universal property:  The image under of a couple in is denoted by, and called a simple tensor. Every element in is a finite sum of such simple tensors.  There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.  In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor product of two Banach spaces and is the "completion" formula_26 of the algebraic tensor product equipped with the projective tensor norm, and similarly for the injective tensor product formula_27. Grothendieck proved in particular that  where is a compact Hausdorff space, the Banach space of continuous functions from to and the space of Bochner-measurable and integrable functions from to, and where the isomorphisms are isometric. The two isomorphisms above are the respective extensions of the map sending the tensor to the vector-valued function.

Tensor products and the approximation property.

Let be a Banach space. The tensor product formula_29 is identified isometrically with the closure in of the set of finite rank operators. When has the approximation property, this closure coincides with the space of compact operators on.  For every Banach space, there is a natural norm linear map  obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when is the dual of. Precisely, for every Banach space, the map  is one-to-one if and only if has the approximation property.  Grothendieck conjectured that formula_26 and formula_27 must be different whenever and are infinite-dimensional Banach spaces. This was disproved by Gilles Pisier in 1983. Pisier constructed an infinite-dimensional Banach space such that formula_34 and formula_35 are equal. Furthermore, just as Enflo's example, this space is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space does not have the approximation property.

Some classification results.

Characterizations of Hilbert space among Banach spaces.

A necessary and sufficient condition for the norm of a Banach space to be associated to an inner product is the parallelogram identity:  It follows, for example, that the Lebesgue space is a Hilbert space only when. If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives:  For complex scalars, defining the inner product so as to be -linear in, antilinear in, the polarization identity gives:  To see that the parallelogram law is sufficient, one observes in the real case that is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and. The parallelogram law implies that is additive in. It follows that it is linear over the rationals, thus linear by continuity.  Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available. The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant : Kwapień proved that if  for every integer and all families of vectors, then the Banach space is isomorphic to a Hilbert space. Here, denotes the average over the possible choices of signs. In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.  Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer, any finite-dimensional normed space, with dimension sufficiently large compared to, contains subspaces nearly isometric to the -dimensional Euclidean space.  The next result gives the solution of the so-called "homogeneous space problem". An infinite-dimensional Banach space is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to is homogeneous, and Banach asked for the converse.  An infinite-dimensional Banach space is hereditarily indecomposable when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces. The Gowers dichotomy theorem asserts that every infinite-dimensional Banach space contains, either a subspace with unconditional basis, or a hereditarily indecomposable subspace, and in particular, is not isomorphic to its closed hyperplanes. If is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and Tomczak–Jaegermann, for spaces with an unconditional basis, that is isomorphic to.

Metric classification.

If formula_40 is an isometry from the Banach space formula_41 onto the Banach space formula_42 (where both formula_41 and formula_42 are vector spaces over formula_45), then the Mazur-Ulam theorem states that formula_46 must be an affine transformation. In particular, if formula_47, this is formula_46 maps the zero of formula_41 to the zero of formula_42, then formula_46 must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.

Topological classification.

Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.  Anderson–Kadec theorem (1965–66) proves that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.

Spaces of continuous functions.

When two compact Hausdorff spaces and are homeomorphic, the Banach spaces and are isometric. Conversely, when is not homeomorphic to, the (multiplicative) Banach–Mazur distance between and must be greater than or equal to, see above the results by Amir and Cambern. Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:  The situation is different for countably infinite compact Hausdorff spaces. Every countably infinite compact is homeomorphic to some closed interval of ordinal numbers  equipped with the order topology, where is a countably infinite ordinal. The Banach space is then isometric to. When are two countably infinite ordinals, and assuming, the spaces and are isomorphic if and only if. For example, the Banach spaces  are mutually non-isomorphic.

Examples.

A glossary of symbols:

Derivatives.

Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateaux derivative for details. The Fréchet derivative allows for an extension of the concept of a directional derivative to Banach spaces. The Gateaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces. Fréchet differentiability is a stronger condition than Gateaux differentiability. The quasi-derivative is another generalization of directional derivative that implies a stronger condition than Gateaux differentiability, but a weaker condition than Fréchet differentiability.

Generalizations.

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions R → R, or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.

In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

Banachovy prostory jsou normované lineární prostory, které jsou navíc úplné. Jsou to jedny z ústředních objektů zkoumání funkcionální analýzy. Jsou pojmenovány podle Stefana Banacha, který je studoval.


Etymology.

The first recorded use of the word "hockey" is in the 1773 book "Juvenile Sports and Pastimes, to Which Are Prefixed, Memoirs of the Author: Including a New Mode of Infant Education" by Richard Johnson (Pseud. Master Michel Angelo), whose chapter XI was titled "New Improvements on the Game of Hockey". The belief that hockey was mentioned in a 1363 proclamation by King Edward III of England is based on modern translations of the proclamation, which was originally in Latin and explicitly forbade the games "Pilam Manualem, Pedivam, & Bacularem: & ad Canibucam & Gallorum Pugnam". The English historian and biographer John Strype did not use the word "hockey" when he translated the proclamation in 1720, instead translating "Canibucam" as "Cambuck"; this may have referred to either an early form of hockey or a game more similar to golf or croquet.  The word "hockey" itself is of unknown origin. One supposition is that it is a derivative of "hoquet", a Middle French word for a shepherd's stave. The curved, or "hooked" ends of the sticks used for hockey would indeed have resembled these staves. Another supposition derives from the known use of cork bungs, (stoppers) in place of wooden balls to play the game. The stoppers came from barrels containing "hock" ale, also called "hocky".

History.

Games played with curved sticks and a ball can be found in the histories of many cultures. In Egypt, 4000-year-old carvings feature teams with sticks and a projectile, hurling dates to before 1272 BC in Ireland, and there is a depiction from approximately 600 BC in Ancient Greece, where the game may have been called "kerētízein" or (κερητίζειν) because it was played with a horn or horn-like stick ("kéras", κέρας). In Inner Mongolia, the Daur people have been playing "beikou", a game similar to modern field hockey, for about 1,000 years.  Most evidence of hockey-like games during the Middle Ages is found in legislation concerning sports and games. The Galway Statute enacted in Ireland in 1527 banned certain types of ball games, including games using "hooked" (written "hockie", similar to "hooky") sticks.  By the 19th century, the various forms and divisions of historic games began to differentiate and coalesce into the individual sports defined today. Organizations dedicated to the codification of rules and regulations began to form, and national and international bodies sprang up to manage domestic and international competition.

Subtypes.

Bandy.

Bandy is played with a ball on a football pitch-sized ice arena (bandy rink), typically outdoors, and with many rules similar to association football. It is played professionally in Russia and Sweden. The sport is recognized by the IOC; its international governing body is the Federation of International Bandy.  Bandy has its roots in England in the 19th century, was originally called "hockey on the ice", and spread from England to other European countries around 1900; a similar Russian sport can also be seen as a predecessor and in Russia, bandy is sometimes called "Russian hockey". Bandy World Championships have been played since 1957 and Women's Bandy World Championships since 2004. There are national club championships in many countries and the top clubs in the world play in the Bandy World Cup every year.

Field hockey.

Field hockey is played on gravel, natural grass, or sand-based or water-based artificial turf, with a small, hard ball approximately 73 mm (2.9 in) in diameter. The game is popular among both males and females in many parts of the world, particularly in Europe, Asia, Australia, New Zealand, South Africa, and Argentina. In most countries, the game is played between single-sex sides, although they can be mixed-sex.  The governing body is the 126-member International Hockey Federation (FIH). Men's field hockey has been played at each Summer Olympic Games since 1908 except for 1912 and 1924, while women's field hockey has been played at the Summer Olympic Games since 1980.  Modern field hockey sticks are constructed of a composite of wood, glass fibre or carbon fibre (sometimes both) and are J-shaped, with a curved hook at the playing end, a flat surface on the playing side and a curved surface on the rear side. All sticks are right-handed – left-handed sticks are not permitted.  While field hockey in its current form appeared in mid-18th century England, primarily in schools, it was not until the first half of the 19th century that it became firmly established. The first club was created in 1849 at Blackheath in south-east London. Field hockey is the national sport of Pakistan. It was the national sport of India until the Ministry of Youth Affairs and Sports declared in August 2012 that India has no national sport.

Ice hockey.

Ice hockey is played between two teams of skaters on a large flat area of ice, using a three-inch-diameter (76.2 mm) vulcanized rubber disc called a puck. This puck is often frozen before high-level games to decrease the amount of bouncing and friction on the ice. The game is played all over North America, Europe and to varying extents in many other countries around the world. It is the most popular sport in Canada, Finland, Latvia, the Czech Republic, and Slovakia. Ice hockey is the national sport of Latvia and the national winter sport of Canada. Ice hockey is played at a number of levels, by all ages.  The governing body of international play is the 77-member International Ice Hockey Federation (IIHF). Men's ice hockey has been played at the Winter Olympics since 1924, and was in the 1920 Summer Olympics. Women's ice hockey was added to the Winter Olympics in 1998. North America's National Hockey League (NHL) is the strongest professional ice hockey league, drawing top ice hockey players from around the globe. The NHL rules are slightly different from those used in Olympic ice hockey over many categories. International ice hockey rules were adopted from Canadian rules in the early 1900s.  The contemporary sport developed in Canada from European and native influences. These included various stick and ball games similar to field hockey, bandy and other games where two teams push a ball or object back and forth with sticks. These were played outdoors on ice under the name "hockey" in England throughout the 19th century, and even earlier under various other names. In Canada, there are 24 reports of hockey-like games in the 19th century before 1875 (five of them using the name "hockey"). The first organized and recorded game of ice hockey was played indoors in Montreal, Quebec, Canada, on March 3, 1875, and featured several McGill University students.  Ice hockey sticks are long L-shaped sticks made of wood, graphite, or composites with a blade at the bottom that can lie flat on the playing surface when the stick is held upright and can legally curve either way, for left- or right-handed players.

Ice sledge hockey.

Ice sledge hockey or para ice hockey is a form of ice hockey designed for players with physical disabilities affecting their lower bodies. Players sit on double-bladed sledges and use two sticks; each stick has a blade at one end and small picks at the other. Players use the sticks to pass, stickhandle and shoot the puck, and to propel their sledges. The rules are very similar to IIHF ice hockey rules.  Canada is a recognized international leader in the development of sledge hockey, and much of the equipment for the sport was first developed there, such as sledge hockey sticks laminated with fiberglass, as well as aluminum shafts with hand-carved insert blades and special aluminum sledges with regulation skate blades.  Based on ice sledge hockey, inline sledge hockey is played to the same rules as inline puck hockey (essentially ice hockey played off-ice using inline skates). There is no classification point system dictating who can play inline sledge hockey, unlike the situation with other team sports such as wheelchair basketball and wheelchair rugby. Inline sledge hockey is being developed to allow everyone, regardless of whether they have a disability or not, to complete up to world championship level based solely on talent and ability. The first game of inline sledge hockey was played at Bisley, England, on 19 December 2009 between the Hull Stingrays and the Grimsby Redwings. Matt Lloyd is credited with inventing inline sledge hockey, and Great Britain is seen as the international leader in the game's development.

Roller hockey (inline).

Inline hockey is a variation of roller hockey very similar to ice hockey, from which it is derived. Inline hockey is played by two teams, consisting of four skaters and one goalie, on a dry rink divided into two halves by a center line, with one net at each end of the rink. The game is played in three 15-minute periods with a variation of the ice hockey off-side rule. Icings are also called, but are usually referred to as illegal clearing. The governing body is the IIHF, as for ice hockey, but some leagues and competitions do not follow the IIHF regulations, in particular USA Inline and Canada Inline.

Roller hockey (quad).

Roller hockey, also known as quad hockey, international-style ball hockey, rink hockey and Hoquei em Patins, is an overarching name for a roller sport that has existed since long before inline skates were invented. This sport is played in over sixty countries and has a worldwide following. Roller hockey was a demonstration sport at the 1992 Barcelona Summer Olympics.

Street hockey.

Also known as road hockey, this is a dry-land variant of ice and roller hockey played year-round on a hard surface (usually asphalt). A ball is usually used instead of a puck, and protective equipment is not usually worn.

Other forms of hockey.

Other games derived from hockey or its predecessors include the following:

Hockey is a sport in which two teams play against each other by trying to manoeuvre a ball or a puck into the opponent's goal using a hockey stick. There are many types of hockey such as bandy, field hockey, ice hockey and rink hockey.

Hokej je kolektivní sport, ve kterém soupeří dva týmy mezi sebou a snaží se dostat puk nebo tvrdý míček do soupeřovy branky pomocí hokejek.


Influence in the Imperial family and the Imperial court.

Distantly related to the imperial family of Constantine, he owed his progression from a less significant Levantine bishopric to the most important episcopal see to his influence at court, and the great power he wielded in the church was derived from that source. In fact, during his time in the imperial court, the Eastern court and the major positions in the Eastern Church were held by Arians or Arian sympathizers. With the exception of a short period of eclipse, he enjoyed the confidence both of Constantine and Constantius II. He also served as the tutor of the later Emperor Julian the Apostate; and it was he who baptized Constantine the Great on May 22, 337 owing to his familial relationship with the emperor. Also during his time in the imperial court, Arianism became more popular with the royal family. It can be logically surmised that Eusebius had a huge hand in the acceptance of Arianism in the Constantinian household. The Arian influence grew so strong during his tenure in the imperial court that it was not until the end of the Constantinian dynasty and the appointment of Theodosius I that Arianism lost its influence in the empire.  It was of particular interest that Eusebius was nearly persecuted because of his close relationship to the Emperor Licinius while serving as bishop of Nicomedia during Licinius' reign.

Relationship with Arius.

Like Arius, he was a pupil of Lucian of Antioch, and it is probable that he held the same views as Arius from the very beginning; he was also one of Arius' most fervent supporters who encouraged Arius. It was also because of this relationship that he was the first person whom Arius contacted after the latter was excommunicated from Alexandria by Alexander I of Alexandria in 321. Apparently, Arius and Eusebius were close enough and Eusebius powerful enough that Arius was able to put his theology down in writing. He afterward modified his ideas somewhat, or perhaps he only yielded to the pressure of circumstances; but he was, if not the teacher, at all events the leader and organizer, of the Arian council.  At the First Council of Nicaea, 325, he signed the Confession, but only after a long and desperate opposition in which he was said to "subscribe with hand only, not heart" according to ancient sources. It was a huge blow to the Arian party since it was surmised that the participants in the First Council of Nicaea were evenly split between non-Arians and Arians. His defense of Arius angered the emperor, and a few months after the council he was sent into exile due to his continual contacts with Arius and his followers. After the lapse of three years, he succeeded in regaining the imperial favor by convincing Constantine that Arius and his views do not conflict with the proclaimed Nicene Creed. After his return in 329 he brought the whole machinery of the state government into action in order to impose his views upon the Church.

Political and religious career.

In complement to his theological interests, Eusebius was a skilled politician. Upon his return, he regained the lost ground resulted from the First Council of Nicaea, established alliances with other groups such as the Melitians and expelled many opponents.  He was described by modern historians as an "ambitious intriguer" and a "consummate political player". He was also described by ancient sources as a high-handed person who was also aggressive in his dealings.; he also used his allies to spy on his opponents.  He was able to dislodge and exile three key opponents who espoused the First Council of Nicaea: Eustathius of Antioch in 330, Athanasius of Alexandria in 335 and Marcellus of Ancyra in 336. This was no small feat since Athanasius was regarded as a "man of God" by Constantine, and both Eustathius and Athanasius held top positions in the church.  Another major feat was his appointment as the Patriarch of Constantinople by expelling Paul I of Constantinople; Paul would eventually return as Patriarch after Eusebius' death.  Even outside the empire, Eusebius had great influence. He brought Ulfilas into the Arian priesthood and sent the latter to convert the heathen Goths.  Eusebius baptised Constantine the Great in his villa in Nicomedia, on May 22, 337 just before the death of the Emperor.

Death and aftermath.

He died at the height of his power in the year 341.  He was so influential that even after his death, Constantius II heeded his and Eudoxus of Constantinople's advice to attempt to convert the Roman Empire to Arianism by creating Arian Councils and official Arian Doctrines.  It was because of Eusebius that "On the whole, Constantine and his successors made life pretty miserable for Church leaders committed to the Nicene decision and its Trinitarian formula."  Eusebius of Nicomedia is not to be confused with his contemporary Eusebius of Caesarea, the author of a well-known early book of Church History.

External links.

Correspondence of Eusebius of Nicomedia:

Eusebius of Nicomedia (died 341) was an Arian priest, the man who baptised Constantine the Great. He was a bishop of Berytus (modern-day Beirut) in Phoenicia. He was later made the bishop of Nicomedia, where the Imperial court resided. He lived finally in Constantinople from 338 up to his death.

Eusebios z Nikomédie († 341) byl jeden z nejvlivnějších křesťanských biskupů první poloviny 4. století a jeden z hlavních představitelů ariánství. Působil v maloasijské Nikomédii, tehdy sídelním městě římských císařů ve východní polovině římské říše.


Formal grammars.

A formal grammar of this type consists of a finite set of "production rules" ("left-hand side" → "right-hand side"), where each side consists of a finite sequence of the following symbols:  A formal grammar provides an axiom schema for (or "generates") a "formal language", which is a (usually infinite) set of finite-length sequences of symbols that may be constructed by applying production rules to another sequence of symbols (which initially contains just the start symbol). A rule may be applied by replacing an occurrence of the symbols on its left-hand side with those that appear on its right-hand side. A sequence of rule applications is called a "derivation". Such a grammar defines the formal language: all words consisting solely of terminal symbols which can be reached by a derivation from the start symbol.  Nonterminals are often represented by uppercase letters, terminals by lowercase letters, and the start symbol by. For example, the grammar with terminals, nonterminals, production rules and start symbol, defines the language of all words of the form formula_1 (i.e. copies of followed by copies of ).  The following is a simpler grammar that defines the same language:  Terminals, Nonterminals, Start symbol, Production rules  As another example, a grammar for a toy subset of English language is given by: and start symbol. An example derivation is Other sequences that can be derived from this grammar are: ""ideas hate great linguists"", and ""ideas generate"". While these sentences are nonsensical, they are syntactically correct. A syntactically incorrect sentence ( e.g. ""ideas ideas great hate"") cannot be derived from this grammar. See "Colorless green ideas sleep furiously" for a similar example given by Chomsky in 1957; see Phrase structure grammar and Phrase structure rules for more natural language examples and the problems of formal grammar in that area.

The hierarchy.

The following table summarizes each of Chomsky's four types of grammars, the class of language it generates, the type of automaton that recognizes it, and the form its rules must have.  Note that the set of grammars corresponding to recursive languages is not a member of this hierarchy; these would be properly between Type-0 and Type-1.  Every regular language is context-free, every context-free language is context-sensitive, every context-sensitive language is recursive and every recursive language is recursively enumerable. These are all proper inclusions, meaning that there exist recursively enumerable languages that are not context-sensitive, context-sensitive languages that are not context-free and context-free languages that are not regular.

Type-0 grammars.

Type-0 grammars include all formal grammars. They generate exactly all languages that can be recognized by a Turing machine. These languages are also known as the "recursively enumerable" or "Turing-recognizable" languages. Note that this is different from the recursive languages, which can be "decided" by an always-halting Turing machine.

Type-1 grammars.

Type-1 grammars generate context-sensitive languages. These grammars have rules of the form formula_2 with formula_3 a nonterminal and formula_4, formula_5 and formula_6 strings of terminals and/or nonterminals. The strings formula_4 and formula_5 may be empty, but formula_6 must be nonempty. The rule formula_10 is allowed if formula_11 does not appear on the right side of any rule. The languages described by these grammars are exactly all languages that can be recognized by a linear bounded automaton (a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.)

Type-2 grammars.

Type-2 grammars generate the context-free languages. These are defined by rules of the form formula_12 with formula_3 being a nonterminal and formula_4 being a string of terminals and/or nonterminals. These languages are exactly all languages that can be recognized by a non-deterministic pushdown automaton. Context-free languages—or rather its subset of deterministic context-free language—are the theoretical basis for the phrase structure of most programming languages, though their syntax also includes context-sensitive name resolution due to declarations and scope. Often a subset of grammars is used to make parsing easier, such as by an LL parser.

Type-3 grammars.

Type-3 grammars generate the regular languages. Such a grammar restricts its rules to a single nonterminal on the left-hand side and a right-hand side consisting of a single terminal, possibly followed by a single nonterminal (right regular). Alternatively, the right-hand side of the grammar can consist of a single terminal, possibly preceded by a single nonterminal (left regular). These generate the same languages. However, if left-regular rules and right-regular rules are combined, the language need no longer be regular. The rule formula_10 is also allowed here if formula_11 does not appear on the right side of any rule. These languages are exactly all languages that can be decided by a finite state automaton. Additionally, this family of formal languages can be obtained by regular expressions. Regular languages are commonly used to define search patterns and the lexical structure of programming languages.

In formal language theory, computer science and linguistics, the Chomsky hierarchy (occasionally referred to as the Chomsky–Schützenberger hierarchy) is a containment hierarchy of classes of formal grammars.

Chomského hierarchie je hierarchie tříd formálních gramatik generujících formální jazyky. Byla vytvořena Noamem Chomskym v roce 1956. Chomského hierarchie se skládá z následujících tříd: