Sequence space

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field "K" of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in "K", and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
The most important sequence spaces in analysis are the ℓ"p" spaces, consisting of the "p"-power summable sequences, with the "p"-norm. These are special cases of L"p" spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted "c" and "c"0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.
Definition.
By definition, a is just an formula_1-valued map formula_2 whose value at formula_3 is denoted by formula_4 instead of the usual parentheses notation formula_5 
Space of all sequences.
Let formula_6 denote the field either of real or complex numbers. The product formula_7 denotes the set of all sequences of scalars in formula_8 
This set can becomes a vector space, called when vector addition is defined by
and the scalar multiplication is defined by 
A sequence space is any linear subspace of formula_11 
 is the space of all sequences valued in formula_12 
Unless indicated otherwise, the space formula_7 is endowed with the product topology, although importantly, its linear subspaces, such as formula_14 for example, are typically endowed with topologies that are from the subspace topology induced by the product topology. The space of all sequences formula_7 (with the product topology) is a Fréchet space, meaning that it is a complete metrizable locally convex topological vector space (TVS). There does not exist any Hausdorff locally convex topology on formula_7 that is strictly coarser than this product topology. 
The space formula_7 is not normable, which means that its topology is can not be defined by any norm. Also, there does not exist continuous norm on formula_11 In fact, as the following theorem shows, whenever formula_1 is a Fréchet space on which there does not exist any continuous norm, then this is due entirely to formula_7 presence as a subspace. 
.</math></li>
</ol>
ℓ"p" spaces.
For formula_21 formula_14 is the subspace of formula_7 consisting of all sequences formula_24 satisfying
If formula_26 then the real-valued operation formula_27 defined by
defines a norm on formula_29 In fact, formula_14 is a complete metric space with respect to this norm, and therefore is a Banach space.
If formula_31 then formula_14 does not carry a norm, but rather a metric defined by
If formula_34 then formula_35 is defined to be the space of all bounded sequences endowed with the norm
formula_35 is also a Banach space.
"c", "c"0 and "c"00.
The space of convergent sequences "c" is a sequence space. This consists of all formula_38 such that lim"n"→∞ "x""n" exists. Since every convergent sequence is bounded, "c" is a linear subspace of It is, moreover, a closed subspace with respect to the infinity norm, and so a Banach space in its own right.
The subspace of null sequences "c"0 consists of all sequences whose limit is zero. This is a closed subspace of "c", and so again a Banach space.
The subspace of eventually zero sequences "c"00 consists of all sequences which have only finitely many nonzero elements. This is not a closed subspace and therefore is not a Banach space with respect to the infinity norm. For example, the sequence formula_39 where formula_40 for the first formula_41 entries (for and is zero everywhere else (i.e. is Cauchy with respect to infinity norm but not convergent (to a sequence in "c"00).
Space ℝ∞ of finite sequences.
Let 
denote where formula_43 denotes the space of all real sequences. 
 formula_44 is defined identically and although this section will focus on formula_45 all definitions and results described below hold for formula_44 just as much as they hold for formula_47 Indeed, if formula_44 is considered as a real vector space, then there exists a real TVS-isomorphism between formula_44 and 
For every natural number let formula_50 denote the usual Euclidean space endowed with the Euclidean topology and let formula_51 denote the canonical inclusion defined by formula_52 so that its image is
and consequently,
Endow the set formula_55 with the final topology formula_56 induced by the family formula_57 of all canonical inclusions, which by definition is the finest topology on formula_55 such that all maps in formula_59 are continuous. 
With this topology, formula_55 becomes a complete Hausdorff locally convex sequential topological vector space that is a Fréchet–Urysohn space. 
The topology formula_56 is strictly finer than the subspace topology induced on formula_55 by formula_63 where formula_43 is endowed with its usual product topology. 
Endow the image formula_65 with the quotient topology induced on it by the bijection formula_66 that is, it is endowed with the Euclidean topology transferred to it from formula_50 via formula_68 
This topology on formula_65 is equal to the subspace topology induced on it by formula_70 
A subset formula_71 is open (resp. closed) in formula_72 if and only if for every formula_73 the set formula_74 is an open (resp. closed) subset of formula_75 
The topology formula_56 is thus coherent with family of subspaces formula_77 
This makes formula_72 into an LB-space. 
Consequently, if formula_79 and formula_80 is a sequence in formula_55 then formula_82 in formula_72 if and only if there exists some formula_3 such that both formula_85 and formula_80 are contained in formula_65 and formula_82 in formula_75 
Often, for every formula_73 the canonical inclusion formula_91 is used to identify formula_50 with its image formula_65 in formula_94 explicitly, the elements formula_95 and formula_96 are identified together. 
Under this identification, formula_97 becomes a direct limit of the direct system formula_98 where for every formula_99 the map formula_100 is the canonical inclusion defined by formula_101 where there are formula_102 trailing zeros.
Other sequence spaces.
The space of bounded series, denote by bs, is the space of sequences formula_103 for which
This space, when equipped with the norm
is a Banach space isometrically isomorphic to formula_106 via the linear mapping
The subspace "cs" consisting of all convergent series is a subspace that goes over to the space "c" under this isomorphism.
The space Φ or formula_108 is defined to be the space of all infinite sequences with only a finite number of non-zero terms (sequences with finite support). This set is dense in many sequence spaces.
Properties of ℓ"p" spaces and the space "c"0.
The space ℓ2 is the only ℓ"p" space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram law
Substituting two distinct unit vectors for "x" and "y" directly shows that the identity is not true unless "p" = 2.
Each ℓ"p" is distinct, in that ℓ"p" is a strict subset of ℓ"s" whenever "p" < "s"; furthermore, ℓ"p" is not linearly isomorphic to ℓ"s" when "p" ≠ "s". In fact, by Pitt's theorem , every bounded linear operator from ℓ"s" to ℓ"p" is compact when "p" < "s". No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of ℓ"s", and is thus said to be strictly singular.
If 1 < "p" < ∞, then the (continuous) dual space of ℓ"p" is isometrically isomorphic to ℓ"q", where "q" is the Hölder conjugate of "p": 1/"p" + 1/"q" = 1. The specific isomorphism associates to an element "x" of ℓ"q" the functional
for "y" in ℓ"p". Hölder's inequality implies that "L""x" is a bounded linear functional on ℓ"p", and in fact
so that the operator norm satisfies
In fact, taking "y" to be the element of ℓ"p" with
gives "L""x"("y") = ||"x"||"q", so that in fact
Conversely, given a bounded linear functional "L" on ℓ"p", the sequence defined by "x""n" = "L"("e""n") lies in ℓ"q". Thus the mapping formula_115 gives an isometry
The map
obtained by composing κ"p" with the inverse of its transpose coincides with the canonical injection of ℓ"q" into its double dual. As a consequence ℓ"q" is a reflexive space. By abuse of notation, it is typical to identify ℓ"q" with the dual of ℓ"p": (ℓ"p")* = ℓ"q". Then reflexivity is understood by the sequence of identifications (ℓ"p")** = (ℓ"q")* = ℓ"p".
The space "c"0 is defined as the space of all sequences converging to zero, with norm identical to ||"x"||∞. It is a closed subspace of ℓ∞, hence a Banach space. The dual of "c"0 is ℓ1; the dual of ℓ1 is ℓ∞. For the case of natural numbers index set, the ℓ"p" and "c"0 are separable, with the sole exception of ℓ∞. The dual of ℓ∞ is the ba space.
The spaces "c"0 and ℓ"p" (for 1 ≤ "p" < ∞) have a canonical unconditional Schauder basis {"e""i" | "i" = 1, 2...}, where "e""i" is the sequence which is zero but for a 1 in the "i" th entry.
The space ℓ1 has the Schur property: In ℓ1, any sequence that is weakly convergent is also strongly convergent . However, since the weak topology on infinite-dimensional spaces is strictly weaker than the strong topology, there are nets in ℓ1 that are weak convergent but not strong convergent.
The ℓ"p" spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ"p" or of "c"0, was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to a quotient space of ℓ1, was answered in the affirmative by . That is, for every separable Banach space "X", there exists a quotient map formula_118, so that "X" is isomorphic to formula_119. In general, ker "Q" is not complemented in ℓ1, that is, there does not exist a subspace "Y" of ℓ1 such that formula_120. In fact, ℓ1 has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take formula_121; since there are uncountably many such "X" 's, and since no ℓ"p" is isomorphic to any other, there are thus uncountably many ker "Q" 's).
Except for the trivial finite-dimensional case, an unusual feature of ℓ"p" is that it is not polynomially reflexive.
ℓ"p" spaces are increasing in "p".
For formula_122, the spaces formula_14 are increasing in formula_124, with the inclusion operator being continuous: for formula_125, one has formula_126.
This follows from defining formula_127 for formula_128, and noting that formula_129 for all formula_130, which can be shown to imply formula_131.
Properties of ℓ1 spaces.
A sequence of elements in ℓ1 converges in the space of complex sequences ℓ1 if and only if it converges weakly in this space. 
If "K" is a subset of this space, then the following are equivalent:
Here "K" being equismall at infinity means that for every formula_132, there exists a natural number formula_133 such that formula_134 for all formula_135.