Locally Convex Spaces

Introduction

A topological vector space is a vector space carrying a topology for which addition and scalar multiplication are continuous; the topology may be defined in a way that is quite far from a distance, and the general theory is correspondingly permissive. Local convexity is the hypothesis that restores the geometry: it says that every point has a fundamental system of neighbourhoods consisting of convex sets, equivalently that the topology is generated by a family of seminorms. Under this single condition the Hahn–Banach theorem holds in its geometric form, the dual space is large enough to separate points, and an extensive and well-behaved duality theory becomes available. Almost every space of interest in analysis is locally convex, and the exceptions — the topological vector spaces whose topology is not generated by any family of seminorms — are the ones in which the geometry has genuinely broken down.

This article develops the foundations of locally convex spaces: seminorms and the topology they generate, the Minkowski gauge of a convex absorbing set, the seminorm characterisation of local convexity, the bounded sets and the Kolmogorov–von Neumann criterion for normability, the metrisable and Fréchet cases, the Hahn–Banach theorem in its analytic and geometric forms, and the classes of barrelled and bornological spaces for which the uniform boundedness principle holds. It fixes the topological vocabulary used by the rest of the Part. The functional-analytic theory of operators, the spectral theorem and the theory of distributions is the subject of Analysis on Linear Spaces in Part III; the theory of the normed and Banach spaces themselves is that of Normed and Banach Spaces, and the theory of topological modules and vector spaces without the convexity hypothesis is that of Topological Modules and Vector Spaces. Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$ and all vector spaces are over $\mathbb{K}$. A topological vector space is a vector space $E$ with a topology making the maps $(x,y) \mapsto x+y$ and $(\lambda,x) \mapsto \lambda x$ continuous; a locally convex space is a topological vector space with a fundamental system of convex neighbourhoods of $0$. The topological dual is written $E'$, and $E^*$ denotes the algebraic dual.

Seminorms and Locally Convex Topologies

Seminorms

Definition. A seminorm on $E$ is a map $p : E \to \mathbb{R}_{\geq 0}$ such that

$$ p(x + y) \leq p(x) + p(y), \qquad p(\lambda x) = \lvert \lambda \rvert p(x) $$

for all $x, y \in E$ and all $\lambda \in \mathbb{K}$. A seminorm is a norm if $p(x) = 0$ implies $x = 0$. The unit ball of a seminorm is $B_p = \{x : p(x) \leq 1\}$, a convex, balanced and absorbing set.

Definition. Let $\mathcal{P}$ be a family of seminorms on $E$. The locally convex topology generated by $\mathcal{P}$ is the topology for which a fundamental system of neighbourhoods of $0$ is given by the finite intersections

$$ U_{p_1, \dots, p_n; \varepsilon} = \{x : p_i(x) < \varepsilon \text{ for } i = 1, \dots, n\} \qquad (\varepsilon > 0), $$

and the topology on $E$ is translation-invariant: a fundamental system at $x_0$ is $x_0 + U$ with $U$ as above. The space is called $(E, \mathcal{P})$.

Theorem. Let $\mathcal{P}$ be a family of seminorms on $E$ and give $E$ the topology generated by $\mathcal{P}$. Then $E$ is a topological vector space; it is Hausdorff if and only if for every $x \neq 0$ there is $p \in \mathcal{P}$ with $p(x) > 0$, equivalently $\bigcap_{p \in \mathcal{P}} \{p = 0\} = \{0\}$; and each $p \in \mathcal{P}$ is continuous.

Proof. The sets $U_{p_1,\dots,p_n;\varepsilon}$ are convex and balanced, and they are closed under finite intersections and under scaling, so they form a fundamental system of neighbourhoods of $0$ for a translation-invariant topology; continuity of addition follows from the subadditivity of each seminorm, and continuity of scalar multiplication from the homogeneity together with the boundedness of scalars on the finite set $p_1, \dots, p_n$; hence the space is a topological vector space. For the Hausdorff criterion, if two distinct points $x \neq y$ are given, choose $p$ with $p(x - y) > 0$ and take the two disjoint neighbourhoods $x + \{p < \varepsilon/2\}$ and $y + \{p < \varepsilon/2\}$; conversely if some $x \neq 0$ has $p(x) = 0$ for all $p$ then every neighbourhood of $0$ contains $x$ and the topology separates no pair $(0,x)$. Continuity of $p$ is the continuity of the maps defining the topology.

Definition. A locally convex space is a topological vector space $E$ in which $0$ has a fundamental system of neighbourhoods consisting of convex sets, equivalently in which every neighbourhood of $0$ contains a convex neighbourhood of $0$; this is a local condition at $0$ because the topology of a topological vector space is translation-invariant. The topology is then a locally convex topology.

Example ($\mathbb{K}^n$ and $\mathbb{K}^I$). On $\mathbb{K}^n$ every norm gives the same topology, generated by the coordinate seminorms $p_i(x) = \lvert x_i \rvert$. On an arbitrary product $\mathbb{K}^I$ the product topology is locally convex, generated by the seminorms $p_i(x) = \lvert x_i \rvert$ for $i \in I$; for infinite $I$ no single norm generates it. The space $\mathbb{K}^{\mathbb{N}}$ with the product topology is complete and metrisable; the space $\mathbb{K}^I$ for uncountable $I$ is not metrisable.

Example (spaces of functions). Let $X$ be a topological space. On the space $C(X)$ of continuous functions the seminorms $p_K(f) = \sup_{x \in K}\lvert f(x) \rvert$, over compact $K \subseteq X$, generate the topology of compact convergence; on the space of continuous functions on a locally compact $X$ this is the topology of uniform convergence on compacta, and it is locally convex. The same construction with the seminorms of the derivatives of every order gives the topology of $C^\infty(X)$ when $X$ is a smooth manifold, which is complete and metrisable in the locally convex sense; the smooth structure and the derivative that define it are Part III's, and that space is treated there. These examples are the reason the theory is developed in this generality: the topologies of analysis are seldom given by a single norm, but they are almost always given by a countable or directed family of seminorms.

The Minkowski Gauge

Definition. Let $A \subseteq E$ be an absorbing set, that is, a set such that for every $x \in E$ there is $t > 0$ with $x \in tA$. The gauge (or Minkowski functional) of $A$ is

$$ p_A(x) = \inf\{t > 0 : x \in tA\} . $$

Proposition. Let $A$ be a convex absorbing set, so that $p_A$ is finite-valued.

(a) If $A$ is also balanced, that is, $\lambda A \subseteq A$ for $\lvert \lambda \rvert \leq 1$, then $p_A$ is a seminorm; conversely, if $A$ is closed and $p_A$ is a seminorm then $A = \{p_A \leq 1\}$ and $A$ is balanced. Closedness is needed: in $E = \mathbb{R}$ the convex absorbing set $A = (-1, 1]$ has gauge $p_A = \lvert \cdot \rvert$, a norm, while $A$ is not balanced and $A \neq \{p_A \leq 1\} = [-1,1]$.

(b) If $A$ is convex, balanced and absorbing, then

$$ \{x : p_A(x) < 1\} \subseteq A \subseteq \{x : p_A(x) \leq 1\} , $$

with equality $A = \{p_A \leq 1\}$ when $A$ is closed and $A = \{p_A < 1\}$ when $A$ is open.

(c) Conversely, if $p$ is a seminorm then $p = p_{B_p}$ with $B_p = \{p \leq 1\}$.

Proof. (a) The homogeneity $p_A(\lambda x) = \lvert \lambda \rvert p_A(x)$ uses the balanced condition for $\lambda$ of modulus at most $1$ and the convexity for the reverse containment; subadditivity uses convexity: if $x \in tA$ and $y \in sA$ then $x + y \in (t+s)A$. For the converse, (b) gives $A \subseteq \{p_A \leq 1\}$; if $p_A(x) = 1$ then $p_A\bigl((1 - \tfrac1n)x\bigr) = 1 - \tfrac1n < 1$, so $(1-\tfrac1n)x \in A$ for every $n$ and $x \in \overline{A} = A$; hence $A = \{p_A \leq 1\}$, a balanced set because $p_A$ is a seminorm. (b) The containments follow from the definition of the infimum: $x \in tA$ with $t<1$ gives $x \in A$ by balancedness and convexity (write $x = t(t^{-1}x) + (1-t)0$), so $p_A(x)<1$ implies $x \in A$, while $x \in A$ gives $p_A(x) \leq 1$ by definition. For the equality cases, $p_A$ is now a seminorm and therefore continuous, so $\{p_A \leq 1\}$ is closed and $\{p_A < 1\}$ is open; if $A$ is closed and $p_A(x) = 1$ then $(1-\tfrac1n)x \in A$ for all $n$ and $x \in \overline{A} = A$, so $A = \{p_A \leq 1\}$; if $A$ is open and $x \in A$, then $(1+\varepsilon)x \in A$ for small $\varepsilon>0$ and $p_A\bigl((1+\varepsilon)x\bigr) \leq 1$, that is, $p_A(x) \leq 1/(1+\varepsilon) < 1$, so $A = \{p_A < 1\}$. (c) is the homogeneity computation.

The Seminorm Characterisation

Theorem (seminorm characterisation). A topological vector space $E$ is locally convex, in the sense that every neighbourhood of $0$ contains a convex neighbourhood of $0$, if and only if its topology is generated by a family of seminorms.

Proof. One direction is the theorem above, which produces a locally convex topology from seminorms. For the other, let $E$ be locally convex and let $\mathcal{U}$ be the family of all open convex balanced neighbourhoods of $0$; the family is a fundamental system of neighbourhoods of $0$ by local convexity and the operation of passing to the convex balanced hull of a convex neighbourhood. For $U \in \mathcal{U}$ the gauge $p_U$ is a seminorm by the proposition, and the sets $\{p_U < 1\} \subseteq U$ show that the topology generated by $\{p_U\}$ is finer than the given one; since each $p_U$ is continuous for the given topology, it is also coarser. Hence the two topologies agree.

Corollary. A locally convex space has a fundamental system of neighbourhoods of $0$ consisting of closed, convex, balanced and absorbing sets; the closed unit balls of the seminorms of a directed generating family form such a system.

Proof. Replacing a convex balanced neighbourhood $U$ of $0$ by its closure preserves convexity, balancedness and the neighbourhood property; the gauges of the resulting family generate the topology, and the family is directed downwards by finite intersections.

Boundedness, Metrisability and Completeness

Bounded Sets

Definition. A subset $B \subseteq E$ of a topological vector space is bounded if for every neighbourhood $U$ of $0$ there is $\lambda > 0$ with $B \subseteq \lambda U$; equivalently, if every continuous seminorm is bounded on $B$. The space $E$ is quasi-complete if every closed and bounded subset is complete, and bornological if every seminorm that is bounded on bounded sets is continuous.

Theorem (Kolmogorov's normability criterion; von Neumann). A Hausdorff locally convex space is normable — that is, its topology is induced by a single norm — if and only if there is a bounded neighbourhood of $0$.

Proof. If the topology is induced by a norm, the unit ball is a bounded neighbourhood of $0$. Conversely let $U$ be a bounded convex balanced neighbourhood of $0$; its gauge $p_U$ is a seminorm, and it is a norm because $U$ is bounded and $E$ is Hausdorff: if $p_U(x) = 0$ then $x \in tU$ for every $t>0$, and boundedness of $U$ forces $x = 0$. The topology induced by $p_U$ is exactly the given topology because $U$ is a neighbourhood of $0$ whose multiples form a fundamental system.

Definition. A metric locally convex space is a locally convex space whose topology is induced by a metric, equivalently by a countable family of seminorms; it is an $F$-space if it is a complete metrisable topological vector space, and a Fréchet space if it is a locally convex $F$-space, that is, a complete metrisable locally convex space.

Theorem (metrisability). A Hausdorff locally convex space $E$ is metrisable if and only if its topology is generated by a countable family of seminorms.

Proof. A countable family $p_1, p_2, \dots$ generates a topology equivalent to that of the metric $d(x,y) = \sum_n 2^{-n}\min(1, p_n(x-y))$, which induces the same neighbourhoods by the standard comparison of a metric with a countable family of seminorms; conversely a metric space has a countable base at $0$ and each neighbourhood of $0$ contains a convex balanced one, whose gauges form a countable generating family.

Example (a metrisable but not normable space). The space $\mathbb{K}^{\mathbb{N}}$ with the product topology is metrisable, its metric being $d(x,y) = \sum_n 2^{-n}\lvert x_n - y_n \rvert/(1 + \lvert x_n - y_n\rvert)$, and it is Fréchet; it is not normable by the von Neumann criterion, because no neighbourhood of $0$ is bounded. The same phenomenon occurs for $C^\infty(X)$ with the seminorms over an increasing exhaustion of $X$ by compact sets.

Completeness and the Hahn–Banach Theorem

Definition. A locally convex space $E$ is complete if every Cauchy net converges; it is sequentially complete if every Cauchy sequence converges. The completion of a Hausdorff locally convex space is a complete Hausdorff locally convex space containing $E$ as a dense subspace, unique up to isomorphism.

Theorem (Hahn–Banach; analytic form). Let $E$ be a real vector space, $p : E \to \mathbb{R}$ a seminorm, $F \subseteq E$ a subspace and $f : F \to \mathbb{R}$ a linear functional with $f(x) \leq p(x)$ for all $x \in F$. Then there is a linear functional $\widetilde{f} : E \to \mathbb{R}$ with $\widetilde{f}\lvert_F = f$ and $\widetilde{f}(x) \leq p(x)$ for all $x \in E$. For a complex vector space the same holds with $\lvert f(x) \rvert \leq p(x)$ on $F$ and $\lvert \widetilde{f}(x) \rvert \leq p(x)$ on $E$.

Proof. The real case proceeds by transfinite extension: the set of pairs $(G, g)$ with $F \subseteq G \subseteq E$ and $g$ linear on $G$ with $g \leq p$ is partially ordered by extension and is inductive, so by Zorn's lemma it has a maximal element; if its domain $G$ were not all of $E$, one adjoins a vector $x_0 \notin G$ and chooses $g(x_0)$ in the interval $[\sup_{y} (-p(-y) + g(y)), \inf_y (p(y) - g(y))]$, which is non-empty by subadditivity of $p$ and linearity of $g$ on $G$; this contradicts maximality. The complex case is reduced to the real one by taking real parts.

Theorem (Hahn–Banach; geometric form). Let $E$ be a locally convex space, let $A, B \subseteq E$ be non-empty disjoint convex sets. If $A$ is open, there is a continuous linear functional $f \in E'$ and a real number $\alpha$ with

$$ f(a) < \alpha \leq f(b) \qquad (a \in A, \ b \in B). $$

If $A$ is compact and $B$ is closed and convex, then the separation is strict, $\sup_A f < \inf_B f$.

Proof. The open case reduces to the analytic form applied to the gauge of the open convex set $A - B$ after translating; the compact case follows by applying the open case to $A + U$ and $-B + U$ for a small neighbourhood $U$ of $0$ and using compactness to obtain strictness. The details are standard.

Corollary (separation of points). Let $E$ be a Hausdorff locally convex space and let $x \in E$, $x \neq 0$. Then there is $f \in E'$ with $f(x) \neq 0$. In particular $E'$ separates points of $E$, and if $f(x) = 0$ for all $f \in E'$ then $x = 0$.

Proof. Apply the geometric form to the disjoint convex sets $A = \{x\}$ and $B = \{0\}$, both closed and convex with $A$ compact.

Remark. The corollary is the reason local convexity is such a powerful hypothesis: the dual space is never too small. In a general topological vector space with no convexity hypothesis the dual may be trivial, and this is the failure that local convexity is designed to avoid. The systematic study of $E'$ with the topologies it carries is the subject, written in this Part; this article uses only the existence of separating functionals.

Barrelled and Bornological Spaces

Barrels and the Uniform Boundedness Principle

Definition. A barrel in a locally convex space $E$ is a closed, convex, balanced and absorbing subset of $E$. The space $E$ is barrelled if every barrel is a neighbourhood of $0$.

Theorem (Banach–Steinhaus; uniform boundedness). Let $E$ be a barrelled locally convex space, $F$ a locally convex space and $\mathcal{F} \subseteq \mathcal{L}(E,F)$ a family of continuous linear maps which is pointwise bounded, that is, $\{T x : T \in \mathcal{F}\}$ is bounded in $F$ for every $x \in E$. Then $\mathcal{F}$ is equicontinuous, that is, for every neighbourhood $V$ of $0$ in $F$ there is a neighbourhood $U$ of $0$ in $E$ with $T(U) \subseteq V$ for all $T \in \mathcal{F}$; in particular $\mathcal{F}$ is bounded in the topology of uniform convergence on a bounded set.

Proof. Let $V$ be a closed convex balanced neighbourhood of $0$ in $F$ and put $A = \bigcap_{T \in \mathcal{F}} T^{-1}(V)$. The set $A$ is closed, convex, balanced and absorbing — absorbing because $\mathcal{F}$ is pointwise bounded and $V$ is a neighbourhood of $0$ — so it is a barrel; by barrelledness it is a neighbourhood of $0$, which is the equicontinuity statement. The boundedness statement follows from the definition of equicontinuity.

Corollary. Every Banach space, and more generally every complete metrisable locally convex space, is barrelled; consequently the uniform boundedness principle holds for families of continuous linear maps on a Fréchet space.

Pro. The Baire category theorem applied to the closed sets $\{x: \sup_T p(Tx) \leq n\}$ shows that some barrel is a neighbourhood of $0$; the argument is the standard Baire-category argument of Banach–Steinhaus (Rudin), the complete metrisable case being the setting.

Bornological Spaces and the Mackey Property

Definition. A locally convex space $E$ is bornological if every seminorm on $E$ which is bounded on the bounded subsets of $E$ is continuous. Every bornological space is barrelled, and every metrisable locally convex space is bornological; these are standard implications, the latter because a metrisable space has a countable base of neighbourhoods of $0$ and the former because a bornological space is an inductive limit of normed spaces, which are barrelled.

Theorem (Mackey–Arens). Let $E$ be a locally convex space with dual $E'$, let $\mathcal{A}$ be a family of weakly bounded subsets of $E'$ covering $E'$, and let $\mathcal{T}_{\mathcal{A}}$ be the topology on $E$ of uniform convergence on the members of $\mathcal{A}$. Then $\mathcal{T}_{\mathcal{A}}$ is compatible with the pairing $\langle E, E'\rangle$ — that is, its continuous dual is exactly $E'$ — if and only if the weak topology and the Mackey topology bound it, $\sigma(E, E') \subseteq \mathcal{T}_{\mathcal{A}} \subseteq \tau(E, E')$, where $\tau(E,E')$ is the topology of uniform convergence on the weakly compact convex balanced subsets of $E'$. The locally convex topologies compatible with the pairing are exactly those between $\sigma(E,E')$ and $\tau(E,E')$, so that $\tau(E,E')$ is the finest topology on $E$ with dual $E'$ and $\sigma(E,E')$ the coarsest.

Pro. The weak topology is generated by the functionals of $E'$ and so has dual $E'$; the Mackey topology is compatible with the pairing by Mackey's theorem. If $\sigma(E,E') \subseteq \mathcal{T} \subseteq \tau(E,E')$ then, passing to duals, $E' = (E,\sigma)' \supseteq (E,\mathcal{T})' \supseteq (E,\tau)' = E'$. Conversely a compatible $\mathcal{T}$ contains $\sigma(E,E')$, the coarsest locally convex topology with dual $E'$, and is contained in $\tau(E,E')$, since no topology compatible with the pairing is finer than the Mackey topology. It is quoted as standard; the systematic treatment is Duality Theory.

Definition. A locally convex space $E$ is a Mackey space if its topology coincides with the Mackey topology $\tau(E, E')$, the finest locally convex topology on $E$ with dual $E'$. Every metrisable locally convex space is a Mackey space, and every bornological space is a Mackey space.

Remark. The classes of spaces introduced here — metrisable, Fréchet, barrelled, bornological, Mackey — are the hypotheses under which the theorems of functional analysis are stated, and the whole is a study of the topologies that the dual carries. This article has introduced the vocabulary; the systematic duality theory follows.

Summary

A topological vector space is a vector space whose topology makes addition and scalar multiplication continuous; it is locally convex when its topology is generated by a family of seminorms, equivalently when $0$ has a fundamental system of convex neighbourhoods. The topology generated by a family $\mathcal{P}$ of seminorms has the finite-intersection neighbourhoods $U_{p_1,\dots,p_n;\varepsilon}$ as a fundamental system; the space is Hausdorff exactly when the seminorms separate points, and the Minkowski gauge $p_A$ of a convex balanced absorbing set $A$ recovers $A$ as its unit ball. By the seminorm characterisation every locally convex topology arises in this way, and the closed unit balls of a directed generating family form a fundamental system of closed convex balanced neighbourhoods.

A set is bounded when every continuous seminorm is bounded on it; by the Kolmogorov–von Neumann criterion a Hausdorff locally convex space is normable exactly when it has a bounded neighbourhood of $0$, and it is metrisable exactly when a countable family of seminorms generates its topology. An $F$-space is a complete metrisable topological vector space and a Fréchet space is a locally convex $F$-space. The Hahn–Banach theorem holds in the analytic form (extension of a functional dominated by a seminorm) and in the geometric form (strict separation of disjoint convex sets, one open or one compact), and it implies that $E'$ separates the points of a Hausdorff locally convex space. A barrel is a closed convex balanced absorbing set; $E$ is barrelled when every barrel is a neighbourhood of $0$, and then the Banach–Steinhaus theorem holds: pointwise bounded families of continuous linear maps are equicontinuous. Every Fréchet space is barrelled and bornological; a bornological space is one whose bounded-set-bounded seminorms are continuous; and Mackey's theorem characterises the spaces for which the dual of a topology of uniform convergence on bounded sets is spanned by the original space.

The vocabulary fixed here — seminorm, gauge, bounded set, normable, metrisable, $F$-space, Fréchet space, barrel, barrelled, bornological, Mackey — is used by the remainder of this category and by the operator-algebraic articles of the next category. The study of the dual space and its topologies is; the theory of the metrisable complete case is; both are written in this Part. The analysis on these spaces belongs to Part III.

Summary of Notation

Symbol	Meaning
$\mathbb{K}$	$\mathbb{R}$ or $\mathbb{C}$
$E$, $F$	Locally convex spaces over $\mathbb{K}$
$p$, $q$	Seminorms
$\mathcal{P}$	A generating family of seminorms
$B_p = \{x : p(x) \leq 1\}$	Unit ball of a seminorm
$p_A$	Minkowski gauge of an absorbing set $A$
$E'$, $E^*$	Topological and algebraic duals
$\mathcal{L}(E,F)$	Continuous linear maps
$U_{p_1,\dots,p_n;\varepsilon}$	Basic convex neighbourhood of $0$
$\mathcal{T}_{\mathcal{A}}$	Topology of uniform convergence on the sets of $\mathcal{A}$
$F$-space	Complete metrisable topological vector space
Fréchet space	Complete metrisable locally convex space