Locally convex topological vector space

A subset C in X is called

1. Convex if for all x, y in C, and 0 ≤ t ≤ 1, tx + (1 – t)y is in C. In other words, C contains all line segments between points in C.
2. Circled if for all x in C, λx is in C if |λ| = 1. If 𝕂 = ℝ, this means that C is equal to its reflection through the origin. For 𝕂 = ℂ, it means for any x in C, C contains the circle through x, centred on the origin, in the one-dimensional complex subspace generated by x.
3. A cone (when the underlying field is ordered) if for all x in C and 0 ≤ λ ≤ 1, λx is in C.
4. Balanced if for all x in C, λx is in C if |λ| ≤ 1. If 𝕂 = ℝ, this means that if x is in C, C contains the line segment between x and −x. For 𝕂 = ℂ, it means for any x in C, C contains the disk with x on its boundary, centred on the origin, in the one-dimensional complex subspace generated by x. Equivalently, a balanced set is a circled cone.
5. Absorbent or absorbing if for every x in X, there exists r > 0 such that x is in tC for all t ∈ 𝕂 satisfying |t| > r. The set C can be scaled out by any "large" value to absorb every point in the space.
   • In any TVS, every neighborhood of the origin is absorbent.[7]
6. Absolutely convex or a disk if it is both balanced and convex. This is equivalent to it being closed under linear combinations whose coefficients absolutely sum to ≤ 1; such a set is absorbent if it spans all of X.

Definition:[7] A topological vector space is called locally convex if the origin has a neighborhood basis (i.e. a local base) consisting of convex sets.

In fact, every locally convex TVS has a neighborhood basis of the origin consisting of absolutely convex sets (i.e. disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets.[7] Every TVS has a neighborhood basis at the origin consisting of balanced sets but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced and convex. Note that it is possible for a TVS to have some neighborhoods of the origin that are convex and yet not be locally convex.

Because translation is (by definition of "topological vector space") continuous, all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.

A seminorm on X is a map p : X → ℝ such that

1. p is positive or positive semidefinite: p(x) ≥ 0;
2. p is positive homogeneous or positive scalable: p(λx) = |λ| p(x) for every scalar λ. So, in particular, p(0) = 0;
3. p is subadditive. It satisfies the triangle inequality: p(x + y) ≤ p(x) + p(y).

If p satisfies positive definiteness, which states that if p(x) = 0 then x = 0, then p is a norm. While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below.

Definition: If X is a vector space and 𝒫 is a family of seminorms on X then a subset 𝒬 of 𝒫 is called a base of seminorms for 𝒫 if for all p ∈ 𝒫 there exists a q ∈ 𝒬 and a real r > 0 such that p ≤ rq.[8]

Definition (second version): A locally convex space is defined to be a vector space X along with a family 𝒫 of seminorms on X.

Suppose that X is a vector space over 𝕂, where 𝕂 is either the real or complex numbers, and let B_<r (resp. B_≤r) denote the open (resp. closed) ball of radius r > 0 in 𝕂. A family of seminorms 𝒫 on the vector space X induces a canonical vector space topology on X, called the initial topology induced by the seminorms, making it into a topological vector space (TVS). By definition, it is the coarsest topology on X for which all maps in 𝒫 are continuous.

That the vector space operations are continuous in this topology follows from properties 2 and 3 above. It can easily be seen that the resulting topological vector space is "locally convex" in the sense of the first definition given above because each U_B, ε (0) is absolutely convex and absorbent (and because the latter properties are preserved by translations).

Note that it is possible for a locally convex topology on a space X to be induced by a family of norms but for X to not be normable (that is, to have its topology be induced by a single norm).

Suppose that 𝒫 is a family of seminorms on X that induces a locally convex topology 𝜏 on X. A subbasis at the origin is given by all sets of the form ${\displaystyle p^{-1}\left(B_{<r}\right)=\left\{x\in X:p(x)<r\right\}}$ as p ranges over 𝒫 and r ranges over the positive real numbers. A base at the origin is given by the collection of all possible finite intersections of such subbasis sets.

Recall that the topology of a TVS is translation invariant, meaning that if S is any subset of X containing the origin then for any x ∈ X, S is a neighborhood of 0 if and only if x + S is a neighborhood of x; thus it suffices to define the topology at the origin. A base of neighborhoods of y for this topology is obtained in the following way: for every finite subset F of 𝒫 and every r > 0, let

${\displaystyle U_{F,r}(y):=\left\{x\in X:p(x-y)<r\ {\text{ for all }}p\in F\right\}}$.

Definition: If X is a locally convex space and if 𝒬 is a collection of continuous seminorms on X, then 𝒬 is called a base of continuous seminorms if it is a base of seminorms for the collection of all continuous seminorms on X.[8]

• Explicitly, this means that for all continuous seminorms p on X, there exists a q ∈ 𝒬 and a real r > 0 such that p ≤ rq.[8]

If 𝒬 is a base of continuous seminorms for a locally convex TVS X then the family of all sets of the form ${\displaystyle \left\{x\in X:q\left(x\right)<r\right\}}$ as q varies over 𝒬 and r varies over the positive real numbers, is a base of neighborhoods of the origin in X (not just a subbasis, so there is no need to take finite intersections of such sets).[8]

Definition: A family 𝒫 of seminorms on a vector space X is called saturated if for any p and q in 𝒫, the seminorm defined by ${\displaystyle x\mapsto \max\{p(x),q(x)\}}$ belongs to 𝒫.

If 𝒫 is a saturated family of continuous seminorms that induces the topology on X then the collection of all sets of the form { x ∈ X : p(x) < r } as p ranges over 𝒫 and r ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets;[8] note that this forms a basis at the origin rather than merely a subbasis so that in particular, there is no need to take finite intersections of such sets.[8]

The following theorem implies that if X is a locally convex space then the topology of X can be a defined by a family of continuous norms on X (a norm is an injective seminorm) if and only if there exists at least one continuous norm on X.[9] If there exists a continuous norm on a topological vector space X then X is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or Fréchet spaces).

Suppose that the topology of a locally convex space X is induced by a family 𝒫 of continuous seminorms on X. If x ∈ X and if x_• = (x_i)_{i ∈ I} is a net in X, then x_• → x in X if and only if for all p ∈ 𝒫, p(x_• - x) = (p(x_i - x))_{i ∈ I} → 0.[10] Moreover, if x_• is Cauchy in X, then so is p(x_•) = (p(x_i))_{i ∈ I} for every p ∈ 𝒫.[10]

Although the definition in terms of a neighborhood base gives a better geometric picture, the definition in terms of seminorms is easier to work with in practice. The equivalence of the two definitions follows from a construction known as the Minkowski functional or Minkowski gauge. The key feature of seminorms which ensures the convexity of their ε-balls is the triangle inequality.

For an absorbing set C such that if x is in C, then tx is in C whenever 0 ≤ t ≤ 1, define the Minkowski functional of C to be

${\displaystyle \mu _{C}(x)=\inf\{\lambda >0:x\in \lambda C\}.}$

From this definition it follows that μ_C is a seminorm if C is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets

${\displaystyle \left\{x:p_{\alpha _{1}}(x)<\varepsilon _{1},\ldots ,p_{\alpha _{n}}(x)<\varepsilon _{n}\right\}}$

form a base of convex absorbent balanced sets.

• Suppose that Y is a TVS (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of Y are exactly those that are of the form z + { y ∈ Y : p(y) < 1 } = { y ∈ Y : p(y - z) < 1} for some z ∈ Y and some positive continuous sublinear functional p on Y.[12]
• If S ⊆ X and x ∈ X, then x ∈ cl S if and only if for every r > 0 and every finite collection p₁, ..., p_n ∈ 𝒫 there exists some s ∈ S such that Σⁿ
  _i=1 p_i(x - s) < r.[13]
• The closure of { 0 } in X is equal to ${\displaystyle \cap _{p\in {\mathcal {P}}}p^{-1}(0)}$.[14]
• Every Hausdorff locally convex TVS is homeomorphic to a subspace of a product of Banach spaces.[15]

• The interior and closure of a convex subset of a TVS is again convex.[16]
• The Minkowski sum of two convex sets is convex; furthermore, the scalar multiple of a convex set is again convex.[16]
• If C is a convex set with non-empty interior, then the closure of C is equal to the closure of the interior of C; furthermore, the interior of C is equal to the interior of the closure of C.[16][17]
  • So if a convex set C has non-empty interior then C is a closed (resp. open) set if and only if it is a regular closed (resp. regular open) set.
• If C is a convex subset of a TVS X (not necessarily Hausdorff), x belongs to the interior of S, and y belongs to the closure of S, then the open line segment joint x and y (i.e. { t x + (1 - t) y : 0 < t < 1}) belongs to the interior of S.[17][18]
• If X is a locally convex space (not necessarily Hausdorff), M is a closed vector subspace of X, V is a convex neighborhood of 0 in M, and if z ∈ X is a vector not in V, then there exists a convex neighborhood U of 0 in X such that V = U ∩ M and z ∉ U.[16]
• The closure of a convex subset of a Hausdorff locally convex TVS X is the same for all locally convex Hausdorff TVS topologies on X that are compatible with duality between X and its continuous dual space.[19]
• In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded.[7]
• In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.[7]
  • More generally, if K is a compact subset of a locally convex space, then the convex hull co K (resp. the disked hull cobal K) is compact if and only if it is complete.[7]
• In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.[20]
• In a Fréchet space, the closed convex hull of a compact set is compact.[21]
• In a locally convex space, any linear combination of totally bounded sets is totally bounded.[20]

For any subset S of a TVS X, the convex hull (resp. closed convex hull, balanced hull, resp. convex balanced hull) of S, denoted by co(S) (resp. ${\displaystyle {\overline {\operatorname {co} }}(S)}$, bal(S), cobal(S)), is the smallest convex (resp. closed convex, balanced, convex balanced) subset of X containing S.

• In a quasi-complete locally convex TVS, the closure of the convex hull of a compact subset is again compact.
• In a Hausdorff locally convex TVS, the convex hull of a precompact set is again precompact.[22] Consequently, in a complete locally convex Hausdorff TVS, the closed convex hull of a compact subset is again compact.[23]
• In any TVS, the convex hull of a finite union of compact convex sets is compact (and convex).[7]
  • Note that this implies that in any Hausdorff TVS, the convex hull of a finite union of compact convex sets is closed (in addition to being compact and convex); in particular, the convex hull of such a union is equal to the closed convex hull of that union.
  • In general, the closed convex hull of a compact set is not necessarily compact.
  • In any non-Hausdorff TVS, there exist subsets that are compact (and thus complete) but not closed.
• The bipolar theorem states that the bipolar (i.e. the polar of the polar) of a subset of a locally convex Hausdorff TVS is equal to the closed convex balanced hull of that set.[24]
• The balanced hull of a convex set is not necessarily convex.
• If C and D are convex subsets of a topological vector space (TVS) X and if x ∈ co(C ∪ D), then there exist c ∈ C, d ∈ D, and a real number r satisfying 0 ≤ r ≤ 1 such that x = r c + (1 - r)d.[16]
• If M is a vector subspace of a TVS X, C a convex subset of M, and D a convex subset of X such that D ∩ M ⊆ C, then C = M ∩ co(C ∪ D).[16]
• Recall that the smallest balanced subset of X containing a set S is called the balanced hull of S and is denoted by bal(S). For any subset S of X, the convex balanced hull of S, denoted by cobal(S), is the smallest subset of X containing S that is convex and balanced.[25] The convex balanced hull of S is equal to the convex hull of the balanced hull of S (i.e. cobal(S) = co(bal(S))), but the convex balanced hull of S is not necessarily equal to the balanced hull of the convex hull of S (i.e. cobal(S) is not necessarily equal to bal(co(S))).[25]
• If A and B are subsets of a TVS X and if r is a scalar then co(A∪B) = co(A)∪co(B), co(rA) = rco(A), and ${\displaystyle {\overline {\operatorname {co} }}(rA)=r{\overline {\operatorname {co} }}(A)}$. Moreover, if ${\displaystyle {\overline {\operatorname {co} }}(A)}$ is compact then ${\displaystyle {\overline {\operatorname {co} }}(A+B)={\overline {\operatorname {co} }}(A)+{\overline {\operatorname {co} }}(B)}$ [26]
• If A and B are subsets of a TVS X whose closed convex hulls are compact, then ${\displaystyle {\overline {\operatorname {co} }}(A\cup B)={\overline {\operatorname {co} }}\left({\overline {\operatorname {co} }}(A)\cup {\overline {\operatorname {co} }}(B)\right)}$.[26]
• If S is a convex set in a complex vector space X and there exists some z ∈ X such that z, iz, -z, -iz ∈ S, then rz + siz ∈ S for all real r, s such that |r| + |s| ≤ 1. In particular, az ∈ S for all scalars a such that |a|² ≤ 1/2.

Coarsest vector topology

Any vector space X endowed with the trivial topology (i.e. the indiscrete topology) is a locally convex TVS (and of course, it is the coarsest such topology). This topology is Hausdorff if and only X = { 0 }. The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS.

In contrast, the discrete topology forms a vector topology on X if and only X = { 0 }. This follows from the fact that every topological vector space is a connected space.

Finest locally convex topology

If X is a real or complex vector space and if 𝒫 is the set of all seminorms on X then the locally convex TVS topology, denoted by 𝜏_lc, that 𝒫 induces on X is called the finest locally convex topology on X.[27] This topology may also be described as the TVS-topology on X having as a neighborhood base at 0 the set of all absorbing disks in X.[27] Any locally convex TVS-topology on X is necessarily a subset of 𝜏_lc. (X, 𝜏_lc) is Hausdorff.[14] Every linear map from (X, 𝜏_lc) into another locally convex TVS is necessarily continuous.[14] In particular, every linear functional on (X, 𝜏_lc) is continuous and every vector subspace of X is closed in (X, 𝜏_lc).;[14] therefore, if X is infinite dimensional then (X, 𝜏_lc) is not pseudometrizable (and thus not metrizable).[27] Moreover, 𝜏_lc is the only Hausdorff locally convex topology on X with the property that any linear map from it into any Hausdorff locally convex space is continuous.[28] The space (X, 𝜏_lc) is a bornological space.[29]

Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalises parts of the theory of normed spaces. The family of seminorms can be taken to be the single norm. Every Banach space is a complete Hausdorff locally convex space, in particular, the L^p spaces with p ≥ 1 are locally convex.

More generally, every Fréchet space is locally convex. A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms.

The space ℝ^ω of real valued sequences with the family of seminorms given by

${\displaystyle p_{i}\left(\left\{x_{n}\right\}_{n}\right)=\left|x_{i}\right|,\qquad i\in \mathbf {N} }$

is locally convex. The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable. Note that this is also the limit topology of the spaces ℝⁿ, embedded in ℝ^ω in the natural way, by completing finite sequences with infinitely many 0.

Given any vector space X and a collection F of linear functionals on it, X can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in F continuous. This is known as the weak topology or the initial topology determined by F. The collection F may be the algebraic dual of X or any other collection. The family of seminorms in this case is given by p_f (x) = | f (x)| for all f in F.

Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions f : ℝⁿ → ℂ such that sup_x|x^aD^bf | < ∞, where a and b are multiindices. The family of seminorms defined by p_a,b( f ) = sup_x |x^aD^bf(x)| is separated, and countable, and the space is complete, so this metrisable space is a Fréchet space. It is known as the Schwartz space, or the space of functions of rapid decrease, and its dual space is the space of tempered distributions.

An important function space in functional analysis is the space D(U) of smooth functions with compact support in U ⊆ ℝⁿ. A more detailed construction is needed for the topology of this space because the space C^∞
₀(U) is not complete in the uniform norm. The topology on D(U) is defined as follows: for any fixed compact set K ⊂ U, the space C^∞
₀(K) of functions f ∈ C^∞
₀(U) with supp( f ) ⊂ K is a Fréchet space with countable family of seminorms || f ||_m = sup_k≤msup_x|D^k f (x)| (these are actually norms, and the completion of the space C^∞
₀(K) with the || ⋅ ||_m norm is a Banach space D^m(K)). Given any collection {K_λ}_λ of compact sets, directed by inclusion and such that their union equal U, the C^∞
₀(K_λ) form a direct system, and D(U) is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More concretely, D(U) is the union of all the C^∞
₀(K_λ) with the strongest locally convex topology which makes each inclusion map C^∞
₀(K_λ) ↪ D(U) continuous. This space is locally convex and complete. However, it is not metrisable, and so it is not a Fréchet space. The dual space of D(ℝⁿ) is the space of distributions on ℝⁿ.

More abstractly, given a topological space X, the space C(X) of continuous (not necessarily bounded) functions on X can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms φ_K( f ) = max{| f (x)| : x ∈ K } (as K varies over the directed set of all compact subsets of X). When X is locally compact (e.g. an open set in ℝⁿ) the Stone-Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of C(X) that separates points and contains the constant functions (e.g., the subalgebra of polynomials) is dense.

Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following:

• The spaces L^p([0, 1]) for 0 < p < 1 are equipped with the F-norm

${\displaystyle \|f\|_{p}^{p}=\int _{0}^{1}|f(x)|^{p}\,dx.}$

They are not locally convex, since the only convex neighborhood of zero is the whole space. More generally the spaces L^p(μ) with an atomless, finite measure μ and 0 < p < 1 are not locally convex.

• The space of measurable functions on the unit interval [0, 1] (where we identify two functions that are equal almost everywhere) has a vector-space topology defined by the translation-invariant metric: (which induces the convergence in measure of measurable functions; for random variables, convergence in measure is convergence in probability)

${\displaystyle d(f,g)=\int _{0}^{1}{\frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}}\,dx.}$

This space is often denoted L₀.

Both examples have the property that any continuous linear map to the real numbers is 0. In particular, their dual space is trivial, that is, it contains only the zero functional.

• The sequence space ℓ^p(N), 0 < p < 1, is not locally convex.

Note that if X is a real or complex vector space, f is a linear functional on X, and p is a seminorm on X, then |f| ≤ p if and only if f ≤ p.[30] If f is a non-0 linear functional on a real vector space X and if p is a seminorm on X, then f ≤ p if and only if ${\displaystyle f^{-1}\left(1\right)\cap \left\{x\in X:p(x)<1\right\}=\emptyset }$.[14]

Let n ≥ 1 be an integer, X₁, ..., X_n be TVSs (not necessarily locally convex), let Y be a locally convex TVS whose topology is determined by a family 𝒬 of continuous seminorms, and let ${\displaystyle M:\prod _{i=1}^{n}X_{i}\to Y}$ be a multilinear operator that is linear in each of its n coordinates. The following are equivalent:

1. M is continuous.
2. For every q ∈ 𝒬, there exist continuous seminorms p₁, ..., p_n on X₁, ..., X_n, respectively, such that ${\displaystyle q\left(M\left(x\right)\right)\leq p_{1}\left(x_{1}\right)\cdots p_{n}\left(x_{n}\right)}$ for all ${\displaystyle x=\left(x_{1},\ldots ,x_{n}\right)\in \prod _{i=1}^{n}X_{i}}$.[14]
3. For every q ∈ 𝒬, there exists some neighborhood of 0 in ${\displaystyle \prod _{i=1}^{n}X_{i}}$ on which q ∘ M is bounded.[14]