Locally convex topological vector space

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.

History

Metrizable topologies on vector spaces have been studied since their introduction in Maurice Frechet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by Felix Hausdorff in 1914,^[1] although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces.^[2]^[3] Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).^[4]^[5]

A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces^[6] (in which case the unit ball of the dual is metrizable).

Definition

Suppose $V$ is a vector space over $K$ , a subfield of the complex numbers (normally $C$ itself or $R$ ). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

Definition via convex sets

A subset $C$ in $V$ is called

Convex if for all $x, y$ in $C$ , and $0 \leq t \leq 1,$ $tx + (1 - t) y$ is in $C$ . In other words, $C$ contains all line segments between points in $C$ .
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 $x$ .
A cone (when the underlying field is ordered) if for all $x$ in $C$ and $0 \leq λ \leq 1,$ $λx$ is in $C$ .
Balanced if for all $x$ in $C$ , $λx$ is in $C$ if $| λ | \leq 1$ . 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.
Absorbent or absorbing if the union of $tC$ over all $t > 0$ is all of $V$ , or equivalently for every $x$ in $V$ , $tx$ is in $C$ for some $t > 0$ . The set $C$ can be scaled out to absorb every point in the space.
Absolutely convex if it is both balanced and convex.

More succinctly, a subset of $V$ is absolutely convex if it is closed under linear combinations whose coefficients absolutely sum to $\leq 1$ . Such a set is absorbent if it spans all of $V$ .

Definition (first version). A topological vector space is called locally convex if the origin has a local base of absolutely convex absorbent sets.

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.

Definition via seminorms

A seminorm on $V$ is a map $p : V \to R$ such that

$p$ is positive or positive semidefinite: $p (x) \geq 0$ .
$p$ is positive homogeneous or positive scalable: $p (λx) = | λ | p (x)$ for every scalar $λ$ . So, in particular, $p (0) = 0$ .
$p$ is subadditive. It satisfies the triangle inequality: $p (x + y) \leq p (x) + p (y)$ .

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 (second version). A locally convex space is defined to be a vector space $V$ along with a family of seminorms ${p α} α \in A$ on $V$ .

A locally convex space carries a natural topology, called the initial topology induced by the seminorms. By definition, it is the coarsest topology for which all the mappings

{\begin{cases}p_{\alpha ,y}:V\to \mathbf {R} \\x\mapsto p_{\alpha }(x-y)&y\in V,\alpha \in A\end{cases}}

are continuous. A base of neighborhoods of $y$ for this topology is obtained in the following way: for every finite subset $B$ of $A$ and every $ε > 0$ , let

U_{B,\varepsilon }(y)=\{x\in V:p_{\alpha }(x-y)<\varepsilon \ \forall \alpha \in B\}.

Note that

U_{B,\varepsilon }(y)=\bigcap _{\alpha \in B}(p_{\alpha ,y})^{-1}([0,\varepsilon )).

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).

Equivalence of definitions

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 $0 \leq t \leq 1$ , define the Minkowski functional of $C$ to be

\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

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

form a base of convex absorbent balanced sets.

Further definitions and properties

A family of seminorms ${p α} α$ is called total or separated or is said to separate points if whenever $p α (x) = 0$ holds for every $α$ then $x$ is necessarily $0$ . A locally convex space is Hausdorff if and only if it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition.
A pseudometric is a generalisation of a metric which does not satisfy the condition that $d (x, y) = 0$ only when $x = y$ . A locally convex space is pseudometrisable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by
$d(x,y)=\sum _{n}^{\infty }{\frac {1}{2^{n}}}{\frac {p_{n}(x-y)}{1+p_{n}(x-y)}}$
(where the $1/2 n$ can be replaced by any positive summable sequence $a n$ ). This pseudometric is translation-invariant, but not homogeneous, meaning $d (kx, ky) \neq | k | d (x, y)$ , and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a Fréchet space.
As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
A Cauchy net in a locally convex space is a net ${x κ} κ$ such that for every $ε > 0$ and every seminorm $p α$ , there exists a $κ$ such that for all $λ, μ > κ$ , $p α (x λ - x μ) < ε$ . In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrisable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space is complete if and only if every Cauchy net converges.
A family of seminorms becomes a preordered set under the relation $p α \leq p β$ if and only if there exists an $M > 0$ such that for all $x$ , $p α (x) \leq Mp β (x)$ . One says it is a directed family of seminorms if the family is a directed set with addition as the join, in other words if for every $α$ and $β$ , there is a $γ$ such that $p α + p β \leq p γ$ . Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family ${p α} α \in I$ , let $Φ$ be the set of finite subsets of $I$ , then for every $F$ in $Φ$ , define
$q_{F}=\sum _{\alpha \in F}p_{\alpha }.$
One may check that ${q F} F \in Φ$ is an equivalent directed family.
If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if $0$ has a bounded neighborhood.

Examples and nonexamples

Examples of locally convex spaces

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 $p \geq 1$ 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

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

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

R n

, embedded in

R ω

in the natural way, by completing finite sequences with infinitely many

0

.

Given any vector space $V$ and a collection $F$ of linear functionals on it, $V$ 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 $V$ 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 $f : R n \to C$ such that $sup x | x a D b f | < \infty$ , where a and b are multiindices. The family of seminorms defined by $p a, b (f) = sup x | x a D b f (x) |$ 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 $U \subseteq R n$ . A more detailed construction is needed for the topology of this space because the space $C \infty 0 (U)$ is not complete in the uniform norm. The topology on $D (U)$ is defined as follows: for any fixed compact set $K \subset U$ , the space $C \infty 0 (K)$ of functions $f \in C \infty 0 (U)$ with $supp(f) \subset K$ is a Fréchet space with countable family of seminorms $‖ f ‖ m = sup x | D m f (x) |$ (these are actually norms, and the space $C \infty 0 (K)$ with the $‖ \cdot ‖ m$ 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 $C \infty 0 (K λ)$ 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 $C \infty 0 (K λ)$ with the strongest locally convex topology which makes each inclusion map $C \infty 0 (K λ) ↪ D (U)$ 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 $D (R n)$ is the space of distributions on $R n$ .
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 $φ K (f) = max{| f (x) | : x \in K}$ (as $K$ varies over the directed set of all compact subsets of $X$ ). When $X$ is locally compact (e.g. an open set in $R n$ ) 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.

Examples of spaces lacking local convexity

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

\|f\|_{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)

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

This space is often denoted

L 0

.

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.

Continuous linear mappings

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally convex spaces are continuous linear maps. Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces.

Given locally convex spaces $V$ and $W$ with families of seminorms ${p α} α$ and ${q β} β$ respectively, a linear map $T : V \to W$ is continuous if and only if for every $β$ , there exist $α 1, α 2, ..., α n$ and $M > 0$ such that for all $v$ in $V$

q_{\beta }(Tv)\leq M\left(p_{\alpha _{1}}(v)+\dotsb +p_{\alpha _{n}}(v)\right).

In other words, each seminorm of the range of $T$ is bounded above by some finite sum of seminorms in the domain. If the family ${p α} α$ is a directed family, and it can always be chosen to be directed as explained above, then the formula becomes even simpler and more familiar:

q_{\beta }(Tv)\leq Mp_{\alpha }(v).

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.

References

Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). Springer. ISBN 0-387-97245-5.
Rudin, Walter (1991). Functional analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5.

^ Hausdorff, F. Grundzüge der Mengenlehre (1914)
^ von Neumann, J. Collected works. Vol II. p.94-104
^ Dieudonne, J. History of Functional Analysis Chapter VIII. Section 1.
^ von Neumann, J. Collected works. Vol II. p.508-527
^ Dieudonne, J. History of Functional Analysis Chapter VIII. Section 2.
^ Banach, S. Theory of linear operations p.75. Ch. VIII. Sec. 3. Theorem 4., translated from Theorie des operations lineaires (1932)

[1] Hausdorff, F. Grundzüge der Mengenlehre (1914)

[2] von Neumann, J. Collected works. Vol II. p.94-104

[3] Dieudonne, J. History of Functional Analysis Chapter VIII. Section 1.

[4] von Neumann, J. Collected works. Vol II. p.508-527

[5] Dieudonne, J. History of Functional Analysis Chapter VIII. Section 2.

[6] Banach, S. Theory of linear operations p.75. Ch. VIII. Sec. 3. Theorem 4., translated from Theorie des operations lineaires (1932)

[1]

[2]

[3]

[4]

[5]

[6]