Complete topological vector space

In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point formula_1 towards which they all get closer to. 
The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point formula_1 towards which they all get closer to" means that this net or filter converges to formula_3 
Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for TVSs, including those that are not metrizable or Hausdorff. 
Completeness is an extremely important property for a topological vector space to possess. 
The notions of completeness for normed spaces and metrizable TVSs, which are commonly defined in terms of completeness of a particular norm or metric, can both be reduced down to this notion of TVS-completeness; a notion that is independent of any particular norm or metric. A metrizable topological vector space formula_4 with a translation invariant metric formula_5 is complete as a TVS if and only if formula_6 is a complete metric space, which by definition means that every formula_5-Cauchy sequence converges to some point in formula_8 
Prominent examples of complete TVSs that are also metrizable include all F-spaces and consequently also all Fréchet spaces, Banach spaces, and Hilbert spaces. 
Prominent examples of complete TVS that are (typically) metrizable include strict LF-spaces and many nuclear spaces such as the Schwartz space of rapidly decreasing smooth functions and also the spaces of distributions and of test functions. 
Explicitly, a topological vector spaces (TVS) is complete if every net, or equivalently, every filter, that is Cauchy with respect to the space's necessarily converges to some point. Said differently, a TVS is complete if its canonical uniformity is a complete uniformity. 
The canonical uniformity on a TVS formula_9 is the unique translation-invariant uniformity that induces on formula_4 the topology formula_11 
This notion of "TVS-completeness" depends on vector subtraction and the topology of the TVS; consequently, it can be applied to all TVSs, including those whose topologies can not be defined in terms metrics or pseudometrics. 
A first-countable TVS is complete if and only if every Cauchy sequence (or equivalently, every elementary Cauchy filter) converges to some point.
Every topological vector space formula_12 even if it is not metrizable or not Hausdorff, has a , which by definition is a complete TVS formula_13 into which formula_4 can be TVS-embedded as a dense vector subspace. Moreover, every Hausdorff TVS has a completion, which is necessarily unique up to TVS-isomorphism. However, as discussed below, all TVSs (even those that are complete, Hausdorff, and/or metrizable) have infinitely many non-Hausdorff completions that are TVS-isomorphic to one another. 
The vector space consisting of scalar-valued simple functions formula_15 for which formula_16 (where this seminorm is defined in the usual way in terms of Lebesgue integration) becomes a seminormed space when endowed with this seminorm, which in turn makes it into both a pseudometric space and a non-Hausdorff non-complete TVS; any completion of this space is a non-Hausdorff complete seminormed space that when quotiented by the closure of its origin (so as to obtain a Hausdorff TVS) results in (a space linearly isometrically-isomorphic to) the usual complete Hausdorff formula_17-space (endowed with the usual complete formula_18 norm). 
As another example demonstrating the usefulness of completions, the completions of topological tensor products, such as projective tensor products or injective tensor products, of the Banach space formula_19 with a complete Hausdorff locally convex TVS formula_20 results in a complete TVS that is TVS-isomorphic to a "generalized" formula_21-space consisting formula_20-valued functions on formula_23 (where this "generalized" TVS is defined analogously to original space formula_19 of scalar-valued functions on formula_23). Similarly, the completion of the injective tensor product of the space of scalar-valued formula_26-test functions with such a TVS formula_20 is TVS-isomorphic to the analogously defined TVS of formula_20-valued formula_26 test functions.
Definitions.
This section summarizes the definition of a complete topological vector space (TVS) in terms of both nets and prefilters. 
Information about convergence of nets and filters, such as definitions and properties, can be found in the article about filters in topology. 
Every topological vector space (TVS) is a commutative topological group with identity under addition and the canonical uniformity of a TVS is defined in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed. 
Canonical uniformity.
The diagonal of formula_4 is the family 
formula_31 
For any formula_32 the canonical entourage/vicinity around formula_33 is the set 
where if formula_35 then formula_36 contains the diagonal 
formula_37 
If formula_33 is symmetric (that is, if formula_39), then formula_36 is , which by definition means that formula_41 and in addition, this symmetric set's with itself is: 
If formula_43 is any neighborhood basis at the origin in formula_9 then the family of subsets of formula_45 : 
is a prefilter on formula_45 . 
If formula_48 is the neighborhood filter at the origin in formula_9 then formula_50 forms a base of entourages for a uniform structure on formula_4 that is considered canonical. 
Explicitly, by definition, formula_9 is the filter formula_53 on formula_45 generated by the above prefilter: 
where formula_56 denotes the of formula_57 in formula_45 . 
If formula_43 is any neighborhood basis at the origin in formula_9 then the filter on formula_45 generated by the prefilter formula_62 is equal to the canonical uniformity formula_53 induced by formula_64 
Let formula_4 and formula_20 be TVSs, formula_67 and formula_68 be a map. Then formula_68 is uniformly continuous if for every neighborhood formula_70 of the origin in formula_12 there exists a neighborhood formula_72 of the origin in formula_20 such that for all formula_74 if formula_75 then formula_76
Cauchy net.
The general theory of uniform spaces has its own definition of a "Cauchy prefilter" and "Cauchy net". For the canonical uniformity on formula_12 these definitions reduces down to the definition described below. 
Suppose formula_78 is a net in formula_4 and formula_80 is a net in formula_81 
The product formula_82 becomes a directed set by declaring formula_83 if and only if formula_84 and formula_85 
Then 
denotes the product net. If formula_87 then the image of this net under the vector addition map formula_88 denotes the sum of these two nets: 
and similarly their difference is defined to be the image of the product net under the vector subtraction map:
A net formula_78 in a TVS formula_4 is called a Cauchy net if 
or equivalently, if for every neighborhood formula_33 of formula_96 in formula_12 there exists some formula_98 such that formula_99 for all formula_100 with formula_101 
A Cauchy sequence is a Cauchy net that is a sequence. 
It suffices to check any of these defining conditions for any given neighborhood basis of formula_96 in formula_8 
Cauchy filter and Cauchy prefilter.
A prefilter formula_104 on an TVS formula_9 called a Cauchy prefilter if it satisfies any of the following equivalent conditions: 
It suffices to check any of the above conditions for any given neighborhood basis of formula_96 in formula_8 
If formula_104 is a prefilter on a TVS formula_4 and formula_110 then formula_111 in formula_4 if and only if formula_113 and formula_104 is Cauchy.
Complete subset.
For any formula_115 a prefilter formula_116 is necessarily a subset of formula_117; that is, formula_118 
A subset formula_23 of a TVS formula_9 is called complete if it satisfies any of the following equivalent conditions: 
The subset formula_23 is called sequentially complete if every Cauchy sequence in formula_23 (or equivalently, every elementary Cauchy filter/prefilter on formula_23) converges to at least one point of formula_124
Importantly, convergence outside of formula_23 is allowed: If formula_4 is not Hausdorff and if every Cauchy prefilter on formula_23 converges to some point of formula_128 then formula_23 will be complete even if some or all Cauchy prefilters on formula_23 converge to points(s) in formula_131 In short, there is no requirement that these Cauchy prefilters on formula_23 converge to points in formula_124 The same can be said of the convergence of Cauchy nets in formula_124 
As a consequence, if a TVS formula_4 is Hausdorff then every subset of the closure of formula_136 in formula_4 is complete because it is compact and every compact set is necessarily complete. In particular, if formula_138 is a proper subset, such as formula_139 for example, then formula_23 would be complete even though Cauchy net in formula_23 (and also every Cauchy prefilter on formula_23) converges to point in formula_143 including those points in formula_144 that do not belong to formula_124 
This example also shows that complete subsets (and indeed, even compact subsets) of a non-Hausdorff TVS may fail to be closed. For example, if formula_138 then formula_147 if and only if formula_23 is closed in formula_8 
Complete topological vector space.
A topological vector space formula_9 is called complete if any of the following equivalent conditions are satisfied:
where if in addition formula_4 is pseudometrizable or metrizable (e.g. a normed space) then this list can be extended to include: 
A topological vector space formula_9 is sequentially complete if any of the following equivalent conditions are satisfied:
Uniqueness of the canonical uniformity.
The existence of the canonical uniformity was demonstrated above by defining it. The theorem below establishes that the canonical uniformity of any TVS formula_9 is the only uniformity on formula_4 that is both (1) translation invariant, and (2) generates on formula_4 the topology formula_11 
This section is dedicated to explaining the precise meanings of the terms involved in this uniqueness statement. 
Uniform spaces and translation-invariant uniformities.
Let formula_115 let formula_158 and formula_159 be subsets of formula_45 , and let formula_161 . Define
where formula_168 (resp. formula_169) is called the set of left (resp. right) formula_158-relatives of (points in) formula_124 The maps formula_172 and formula_173 are the canonical projections onto the first and second coordinates, respectively. 
Given formula_174 formula_158 is symmetric if formula_176 while a subset formula_177 is called formula_158-small if formula_179 Two points formula_1 and formula_181 are formula_158-close if formula_183 
If formula_184 and formula_185 then:
A non-empty family formula_186 is called a base or fundamental system of entourages if formula_104 is a prefilter on formula_45 satisfying all of the following conditions:
A uniformity or uniform structure on formula_4 is a filter formula_190 on formula_45 that is generated by some base of entourages formula_192 in which case we say that formula_104 is a base of entourages . 
For a commutative additive group formula_12 a fundamental system of entourages formula_104 is called translation-invariant if for every formula_196 formula_197 if and only if formula_198 for all formula_199 A uniformity formula_104 is called translation-invariant if it has a base of entourages that is translation-invariant. The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin. 
The canonical uniformity on any TVS is translation-invariant. 
Topology generated by a uniformity.
Let formula_186 be a base of entourages on formula_8 For every formula_177 and formula_161 , the neighborhood prefilter on formula_23 (resp. at formula_206) is the set
and the filter on formula_4 that it generates is known as the neighborhood filter of formula_23 (resp. of formula_206). 
The assignment
of points to prefilters generates a topology on formula_4 called the topology induced by formula_104. 
A subset formula_216 is open in this topology if and only if any of the following equivalent conditions hold:
The closure of a subset formula_177 in this topology is: 
A prefilter formula_219 on a uniform space formula_4 with uniformity formula_190 is called a Cauchy prefilter if for every entourages formula_222 there exists some formula_223such that formula_224
A uniform space formula_225 is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on formula_4 converges to at least one point of formula_4 when formula_4 is endowed with the topology induced by formula_229
If formula_9 is a TVS then for any formula_177 and formula_232 
The topology induced on formula_4 by the canonical uniformity is the same as the topology that formula_4 started with (i.e. it is formula_237).
TVS completeness vs completeness of (pseudo)metrics.
Preliminaries: Complete pseudometric spaces.
We review the basic notions related to the general theory of complete pseudometric spaces. 
Recall that every metric is a pseudometric and that a pseudometric formula_206 is a metric if and only if formula_239 implies formula_240 Thus every metric space is a pseudometric space and a pseudometric space formula_241 is a metric space if and only if formula_206 is a metric. 
If formula_23 is a subset of a pseudometric space formula_6 then the diameter of formula_23 is defined to be 
A prefilter formula_104 on a pseudometric space formula_6 is called a formula_5-Cauchy prefilter or simply a Cauchy prefilter if for each real formula_250 there is some formula_251 such that the diameter of formula_252 is less than formula_253 
Suppose formula_6 is a pseudometric space. A net formula_78 in formula_4 is called a formula_5-Cauchy net or simply a Cauchy net if formula_258 is a Cauchy prefilter, which happens if and only if 
A Cauchy sequence is a sequence that is also a Cauchy net.
Every pseudometric formula_206 on a set formula_4 induces the usual canonical topology on formula_12 which we'll denote by formula_268; it also induces a canonical uniformity on formula_12 which we'll denote byformula_270 The topology on formula_4 induced by the uniformityformula_272 is equal to formula_273 A net formula_78 in formula_4 is Cauchy with respect to formula_206 if and only if it is Cauchy with respect to the uniformityformula_270 
The pseudometric space formula_241 is a complete (resp. a sequentially complete) pseudometric space if and only if formula_279 is a complete (resp. a sequentially complete) uniform space. 
Moreover, the pseudometric space formula_241 (resp. the uniform space formula_279) is complete if and only if it is sequentially complete. 
A pseudometric space formula_6 (e.g. a metric space) is called complete and formula_5 is called a complete pseudometric if any of the following equivalent conditions hold:
And if addition formula_5 is a metric then we may add to this list:
Complete pseudometrics and complete TVSs.
Every F-space, and thus also every Fréchet space, Banach space, and Hilbert space is a complete TVS. Note that every "F"-space is a Baire space but there are normed spaces that are Baire but not Banach. 
A pseudometric formula_5 on a vector space formula_4 is said to be translation invariant if formula_290 for all vectors formula_199
Suppose formula_9 is pseudometrizable TVS (e.g. a metrizable TVS) and that formula_206 is pseudometric on formula_4 such that the topology on formula_4 induced by formula_206 is equal to formula_11 
If formula_206 is translation-invariant, then formula_9 is a complete TVS if and only if formula_241 is a complete pseudometric space. 
If formula_206 is translation-invariant, then may be possible for formula_9 to be a complete TVS but formula_241 to be a complete pseudometric space (see this footnote for an example). 
Complete norms and equivalent norms.
Two norms on a vector space are called equivalent if and only if they induce the same topology. If formula_206 and formula_305 are two equivalent norms on a vector space formula_4 then the normed space formula_241 is a Banach space if and only if formula_308 is a Banach space. 
See this footnote for an example of a continuous norm on a Banach space that is not is equivalent to that Banach space's given norm. 
All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space. Every Banach space is a complete TVS. A normed space is a Banach space (that is, its canonical norm-induced metric is complete) if and only if it is complete as a topological vector space.
Completions.
A completion of a TVS formula_4 is a complete TVS that contains a dense vector subspace that is TVS-isomorphic to formula_8 
Non-uniqueness of all completions.
As the example below shows, regardless of whether or not a space is Hausdorff or already complete, every topological vector space (TVS) has infinitely many non-isomorphic completions. 
However, every Hausdorff TVS has a completion that is unique up to TVS-isomorphism. But nevertheless, every Hausdorff TVS still has infinitely many non-isomorphic non-Hausdorff completions. 
Example (Non-uniqueness of completions): 
Let formula_13 denote any complete TVS and let formula_312 denote any TVS endowed with the indiscrete topology, which recall makes formula_312 into a complete TVS. 
Since both formula_312 and formula_13 are complete TVSs, so is their product formula_316 
If formula_70 and formula_72 are non-empty open subsets of formula_312 and formula_320 respectively, then formula_321 and formula_322 which shows that formula_323 is a dense subspace of formula_316 
Thus by definition of "completion," formula_325 is a completion of formula_323 (it doesn't matter that formula_323 is already complete). 
So by identifying formula_323 with formula_320 if formula_330 is a dense vector subspace of formula_320 then formula_4 has both formula_13 and formula_325 as completions. 
Hausdorff completions.
Every Hausdorff TVS has a completion that is unique up to TVS-isomorphism. But nevertheless, as shown above, every Hausdorff TVS still has infinitely many non-isomorphic non-Hausdorff completions. 
A Cauchy filter formula_104 on a TVS formula_4 is called a minimal Cauchy filter if there does exist a Cauchy filter on formula_4 that is strictly coarser than formula_104 (that is, "strictly coarser than formula_104" means contained as a proper subset of formula_104). 
If formula_104 is a Cauchy filter on formula_4 then the filter generated by the following prefilter:
is the unique minimal Cauchy filter on formula_4 that is contained as a subset of formula_345 
In particular, for any formula_232 the neighborhood filter at formula_1 is a minimal Cauchy filter. 
Let formula_348 be the set of all minimal Cauchy filters on formula_4 and let formula_350 be the map defined by sending formula_110 to the neighborhood filter of formula_1 in formula_8 
We endow formula_348 with the following vector space structure. Given formula_355 and a scalar formula_356 let formula_357 (resp. formula_358 denote the unique minimal Cauchy filter contained in the filter generated by formula_359 (resp. formula_360). 
For every balanced neighborhood formula_33 of the origin in formula_12 let
If formula_4 is Hausdorff then the collection of all sets formula_365 as formula_33 ranges over all balanced neighborhoods of the origin in formula_12 forms a vector topology on formula_348 making formula_348 into a complete Hausdorff TVS. Moreover, the map formula_350 is a TVS-embedding onto a dense vector subspace of formula_371 
If formula_4 is a metrizable TVS then a Hausdorff completion of formula_4 can be constructed using equivalence classes of Cauchy sequences instead of minimal Cauchy filters.
Non-Hausdorff completions.
We now show how every non-Hausdorff TVS formula_4 can be TVS-embedded onto a dense vector subspace of a complete TVS. The proof that every Hausdorff TVS has a Hausdorff completion is widely available so we use its conclusion to prove that every non-Hausdorff TVS also has a completion. These details are sometimes useful for extending results from Hausdorff TVSs to non-Hausdorff TVSs. 
Let formula_375 denote the closure of the origin in formula_12 where formula_312 is endowed with its subspace topology induced by formula_4 (so that formula_312 has the indiscrete topology). 
Since formula_312 has the trivial topology, it is easily shown that every vector subspace of formula_4 that is an algebraic complement of formula_312 in formula_4 is necessarily a topological complement of formula_312 in formula_8 
Let formula_386 denote any topological complement of formula_312 in formula_12 which is necessarily a Hausdorff TVS (since it is TVS-isomorphic to the quotient TVS formula_389). 
Since formula_4 is the topological direct sum of formula_312 and formula_386 (which means that formula_393 in the category of TVSs), the canonical map 
is a TVS-isomorphism. 
Let formula_396 denote the inverse of this canonical map. (As a side note, it follows that every open and every closed subset formula_70 of formula_4 satisfies formula_399)
The Hausdorff TVS formula_386 can be TVS-embedded, say via the map formula_401 onto a dense vector subspace of its completion formula_402 
Since formula_312 and formula_13 are complete, so is their product formula_316 
Let formula_406 denote the identity map and observe that the product map formula_407 is a TVS-embedding whose image is dense in formula_316 
Define the map
which is a TVS-embedding of formula_393 onto a dense vector subspace of the complete TVS formula_316 
Moreover, observe that the closure of the origin in formula_325 is equal to formula_414 and that formula_415 and formula_323 are topological complements in formula_316 
To summarize, given any algebraic (and thus topological) complement formula_386 of formula_419 in formula_4 and given any completion formula_13 of the Hausdorff TVS formula_386 such that formula_423 then the natural inclusion
is a well-defined TVS-embedding of formula_4 onto a dense vector subspace of the complete TVS formula_426 where moreover, we have 
Topology of a completion.
Said differently, if formula_13 is a completion of a TVS formula_4 with formula_330 and if formula_431 is a neighborhood base of the origin in formula_12 then the family of sets 
is a neighborhood basis at the origin in formula_402
Properties preserved by completions.
If a TVS formula_4 has any of the following properties then so does its completion:
If formula_4 is a Hausdorff TVS, then the continuous dual space of formula_4 is identical to the continuous dual space of the completion of formula_8 The completion of a locally convex bornological space is a barrelled space. If formula_4 and formula_20 are DF-spaces then the projective tensor product, as well as its completion, of these spaces is a DF-space. 
The completion of the projective tensor product of two nuclear spaces is nuclear. The completion of a nuclear space is TVS-isomorphic with a projective limit of Hilbert spaces.
Properties of maps preserved by extensions to a completion.
If formula_441 is a nuclear linear operator between two locally convex spaces and if formula_13 be a completion of formula_4 then formula_15 has a unique continuous linear extension to a nuclear linear operator formula_445 
Let formula_4 and formula_20 be two Hausdorff TVSs with formula_20 complete. Let formula_13 be a completion of formula_8 Let formula_451 denote the vector space of continuous linear operators and let formula_452 denote the map that sends every formula_453to its unique continuous linear extension on formula_402 Then formula_452 is a (surjective) vector space isomorphism. Moreover, formula_452 maps families of equicontinuous subsets onto each other. Suppose that formula_451 is endowed with a formula_458-topology and that formula_459 denotes the closures in formula_13 of sets in formula_461 Then the map formula_462 is also a TVS-isomorphism. 
Properties.
Complete TVSs.
Every TVS has a completion and every Hausdorff TVS has a Hausdorff completion. Every complete TVS is quasi-complete space and sequentially complete. However, the converses of the above implications are generally false. There exists a sequentially complete locally convex TVS that is not quasi-complete.
If a TVS has a complete neighborhood of the origin then it is complete. 
Every complete metrizable TVS is a barrelled space and a Baire space (and thus non-meager). The dimension of a complete metrizable TVS is either finite or uncountable.
Cauchy nets and prefilters.
Any neighborhood basis of any point in a TVS is a Cauchy prefilter.
Every convergent net (resp. prefilter) in a TVS is necessarily a Cauchy net (resp. a Cauchy prefilter). Any prefilter that is subordinate to (i.e. finer than) a Cauchy prefilter is necessarily also a Cauchy prefilter and any prefilter finer than a Cauchy prefilter is also a Cauchy prefilter. The filter associated with a sequence in a TVS is Cauchy if and only if the sequence is a Cauchy sequence. Every convergent prefilter is a Cauchy prefilter. 
If formula_4 is a TVS and if formula_110 is a cluster point of a Cauchy net (resp. Cauchy prefilter), then that Cauchy net (resp. that Cauchy prefilter) converges to formula_1 in formula_8 
If a Cauchy filter in a TVS has an accumulation point formula_1 then it converges to formula_3 
Uniformly continuous maps send Cauchy nets to Cauchy nets. A Cauchy sequence in a Hausdorff TVS formula_4 is not necessarily relatively compact (i.e. its closure in formula_4 is not necessarily compact). 
Suppose that formula_471 is a family of TVSs and that formula_4 denotes the product of these TVSs. Suppose that for every index formula_473 formula_474is a prefilter on formula_475 Then the product of this family of prefilters is a Cauchy filter on formula_4 if and only if each formula_474is a Cauchy filter on formula_475
Maps.
If formula_441 is an injective topological homomorphism from a complete TVS into a Hausdorff TVS then the image of formula_15 (that is, formula_481) is a closed subspace of formula_81 
If formula_441 is a topological homomorphism from a complete metrizable TVS into a Hausdorff TVS then the range of formula_15 is a closed subspace of formula_81 
If formula_441 is a uniformly continuous map between two Hausdorff TVSs then the image under formula_15 of a totally bounded subset of formula_4 is a totally bounded subset of formula_81 
Suppose that formula_68 is a uniformly continuous map from a dense subset formula_491 of a TVS formula_4 into a complete Hausdorff TVS formula_81 Then formula_15 has a unique uniformly continuous extension to all of formula_8 If in addition formula_15 is a homomorphism then its unique uniformly continuous extension is also a homomorphism. This remains true if "TVS" is replaced by "commutative topological group." The map formula_15 is not required to be a linear map and that formula_491 is not required to be a vector subspace of formula_8
Suppose formula_441 be a continuous linear operator between two Hausdorff TVSs. If formula_501 is a dense vector subspace of formula_4 and if the restriction formula_503 to formula_501 is a topological homomorphism then formula_441 is also a topological homomorphism. So if formula_13 and formula_491 are Hausdorff completions of formula_4 and formula_509 respectively, and if formula_441 is a topological homomorphism, then formula_15's unique continuous linear extension formula_512 is a topological homomorphism. (Note that it's possible for formula_441 to be surjective but for formula_512 to be injective.)
Suppose formula_4 and formula_20 are Hausdorff TVSs, formula_501 is a dense vector subspace of formula_12 and formula_33 is a dense vector subspaces of formula_81 If formula_501 are and formula_33 are topologically isomorphic additive subgroups via a topological homomorphism formula_15 then the same is true of formula_4 and formula_20 via the unique uniformly continuous extension of formula_15 (which is also a homeomorphism).
Subsets.
Every complete subset of a TVS is sequentially complete. A complete subset of a Hausdorff TVS formula_4 is a closed subset of formula_8 
Every compact subset of a TVS is complete (even if the TVS is not Hausdorff or not complete). Closed subsets of a complete TVS are complete; however, if a TVS formula_4 is not complete then formula_4 is a closed subset of formula_4 that is not complete. The empty set is complete subset of every TVS. If formula_13 is a complete subset of a TVS (the TVS is not necessarily Hausdorff or complete) then any subset of formula_13 that is closed in formula_13 is complete.
If formula_4 is a non-normable Fréchet space on which there exists a continuous norm then formula_4 contains a closed vector subspace that has no topological complement. If formula_4 is a complete TVS and formula_501 is a closed vector subspace of formula_4 such that formula_540 is not complete, then formula_386 does have a topological complement in formula_8 
Let formula_501 be a separable locally convex metrizable topological vector space and let formula_13 be its completion. If formula_23 is a bounded subset of formula_13 then there exists a bounded subset formula_547 of formula_4 such that formula_549
A subset of a TVS ( assumed to be Hausdorff or complete) is compact if and only if it is complete and totally bounded. Thus a closed and totally bounded subset of a complete TVS is compact.
Every complete totally bounded set is relatively compact. 
If formula_4 is any TVS then the quotient map formula_551 is a closed map and thus formula_552 A subset formula_23 of a TVS formula_4 is totally bounded if and only if its image under the canonical quotient map formula_551 is totally bounded. Thus formula_23 is totally bounded if and only if formula_557 is totally bounded. In any TVS, the closure of a totally bounded subset is again totally bounded. In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded. If formula_23 is a subset of a TVS formula_4 such that every sequence in formula_23 has a cluster point in formula_23 then formula_23 is totally bounded. A subset formula_23 of a Hausdorff TVS formula_4 is totally bounded if and only if every ultrafilter on formula_23 is Cauchy, which happens if and only if it is pre-compact (i.e. its closure in the completion of formula_4 is compact). 
If formula_177 is compact, then formula_568 and this set is compact. Thus the closure of a compact set is compact (that is, all compact sets are relatively compact). Thus the closure of a compact set is compact. Every relatively compact subset of a Hausdorff TVS is totally bounded. 
In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact. More generally, if formula_569 is a compact subset of a locally convex space, then the convex hull formula_570 (resp. the disked hull formula_571) is compact if and only if it is complete. Every subset formula_23 of formula_144 is compact and thus complete. In particular, if formula_4 is not Hausdorff then there exist compact complete sets that are not closed. 