The main objective of this chapter is to present the definition of topological vector spaces and to derive some fundamental properties. We will also introduce dual pairs of vector spaces and the weak topology. We start the chapter by briefly recalling concepts of topology and continuity, thereby also fixing notation.

Let X be a set, \(\tau \subseteq \mathcal P(X)\) (the power set of X). Then τ is called a topology , and (X, τ) is called a topological space , if

  • for any \(\mathcal S\subseteq \tau \) one has ,

  • for any finite \(\mathcal F\subseteq \tau \) one has .

(This definition is with the understanding that \(\bigcup \varnothing =\varnothing \), \(\bigcap \varnothing =X\), with the consequence that always .) Concerning notation, we could also write

If or are families of sets, with N finite, then one can also write

The sets are called open, whereas a set A ⊆ X is called closed if X ∖ A is open. For a set B ⊆ X we define

  • , the interior of B (an open set),

  • , the closure of B (a closed set).

For , a set U ⊆ X is called a neighbourhood of x if , and the collection

is called the neighbourhood filter of x. (Note that if .) A neighbourhood base \(\mathcal B\) of x is a collection with the property that the neighbourhood filter coincides with the collection of supersets of sets in \(\mathcal B\). (Note that neighbourhoods need not be open sets.)

A topological space (X, τ) is called Hausdorff if for any , x ≠ y, there exist neighbourhoods U of x, V  of y such that \(U\cap V=\varnothing \).

If (X, d) is a semi-metric space , i.e., X is a set and the semi-metric d: X × X → [0, ∞) is symmetric and satisfies d(x, x) = 0 () as well as the triangle inequality

then d induces a topology τ d on X: A set U ⊆ X is defined to be open if for all there exists r > 0 such that B(x, r) ⊆ U, where

is the open ball with centre x and radius r. The corresponding closed ball will be denoted by

(We mention that our definition of ‘semi-metric’ often runs under the name ‘pseudo-metric’; we found our notation more convenient, as it is parallel to ‘semi-norm’, mentioned later.) The topology τ d is Hausdorff if and only if d is a metric, i.e., additionally to the previous properties one has that d(x, y) = 0 implies x = y.

A topological space (X, τ) is called (semi-)metrisable if there exists a (semi-)metric on X such that τ = τ d.

If τ ⊇ σ are topologies on a set X, then τ is said to be finer (or stronger) than σ, and σ is said to be coarser (or weaker) than τ. The trivial topology \(\{\varnothing ,X\}\) is the coarsest topology on X, and the discrete topology \(\mathcal P(X)\), i.e., the collection of all subsets of X, is the finest topology on X.

Let (X, τ), (Y, σ) be topological spaces, f : X → Y , . Then f is continuous atx if f −1(V ) is a neighbourhood of x, for all neighbourhoods V  of f(x). The mapping f is called continuous , if f is continuous at every , and this is equivalent to the property that for all . The mapping f is a homeomorphism , if f is continuous and bijective, and the inverse f −1: Y → X is also continuous.

FormalPara Remark 1.1

Let X be a set, \(\Gamma \subseteq \mathcal P(\mathcal P(X))\) a set of topologies. Then it is easy to see that \(\bigcap \Gamma \) is a topology on X. In order to spell this out more explicitly, we note that

(In this case, because of the subscript ‘’, \(\bigcap \tau \) does not mean .) △

Let X be a set, \(\mathcal S\subseteq \mathcal P(X)\). Then

is the coarsest topology containing \(\mathcal S\), called the topology generated by \(\mathcal S\), and \(\mathcal S\) is called a subbase of \( \mathop {\mathrm {top}} \nolimits \mathcal S\).

If τ is a topology, \(\mathcal B\subseteq \tau \), and for all one has that

then \(\mathcal B\) is called a base for τ. If \(\mathcal S\) is a subbase of τ, then it is not difficult to show that

(1.1)

is a base of τ.

Let X be a set. Let I be an index set (i.e., a set whose elements we use as indices), and for let (X ι, τ ι) be a topological space and f ι: X → X ι a mapping. The topology

(1.2)

is the coarsest topology on X for which all mappings f ι are continuous; it is called the initial topology with respect to the family . A base of the initial topology is given by

(1.3)

this is a consequence of (1.1) and (1.2).

The product topology on is the initial topology with respect to the family of the canonical projections. A base of the product topology is given by

The following theorem is an important key result on initial topologies, which will be used repeatedly in this treatise.

FormalPara Theorem 1.2

Let (Y, σ), (X, τ), (X ι, τ ι) be topological spaces, g: Y → X, f ι: X → X ι , τ the initial topology with respect to . Let . Then:

  1. (a)

    g is continuous at y if and only if f ι ∘ g is continuous at y .

  2. (b)

    g is continuous if and only if f ι ∘ g is continuous .

  3. (c)

    The initial topology on Y  with respect to g is the same as the initial topology with respect to .

FormalPara Proof

(a) The necessity is clear. In order to show the sufficiency, let U be a neighbourhood of g(y). There exist a finite set F ⊆ I and such that is a neighbourhood of g(y). (Recall that these sets constitute a base of the initial topology.) Therefore, the set

is a neighbourhood of y, and is a subset of g −1(U).

(b) is a consequence of (a).

(c) is an immediate consequence of (b). □

Next we define topological vector spaces and derive some basic properties.

Let E be a vector space over the field (where ), and let τ be a topology on E. Then τ is called a linear topology, and (E, τ) is called a topological vector space , if the mappings

  • a: E × E → E, (x, y)↦x + y,

  • , (λ, x)↦λx

are continuous.

In a topological vector space (E, τ) we will denote the neighbourhood filter of zero by (or , or ).

FormalPara Examples 1.3

(a) A vector space E with the trivial topology \(\tau =\{\varnothing ,E\}\) is a topological vector space.

(b) A vector space E ≠ {0} with the discrete topology is not a topological vector space. Indeed, it is easy to see that the scalar multiplication m is not continuous.

(c) The scalars \(\mathbb R\) and are topological vector spaces.

(d) Normed and semi-normed spaces are topological vector spaces. △

For more explanation on Example 1.3(d) we recall that a semi-norm p on a vector space E is a mapping p: E → [0, ∞) satisfying

  • \(p(x+y)\leqslant p(x)+p(y)\) (), the triangle inequality ,

  • p(λx) = |λ|p(x) (), i.e., p is absolutely homogeneous .

The semi-norm p gives rise to a semi-metric d on E, defined by d(x, y) := p(x − y) (). Then the inequalities \(p((x+y)-(x_0-y_0))\leqslant p(x-x_0)+p(y-y_0)\) and \(p(\lambda x-\lambda _0 x_0)\leqslant |\lambda |p(x-x_0)+|\lambda -\lambda _0|p(x_0)\) () show the continuity of addition and scalar multiplication. The semi-metric d is a metric if and only if p is a norm , i.e., if additionally p(x) = 0 implies x = 0, for .

In the following theorem we collect some basic properties of topological vector spaces.

FormalPara Theorem 1.4

Let (E, τ) be a topological vector space. Then:

  1. (a)

    For all the mapping a x: E → E, y↦x + y is a homeomorphism. The topology τ is determined by a neighbourhood base of zero.

  2. (b)

    For all the mapping m λ: E → E, x↦λx is a homeomorphism.

  3. (c)

    Each isabsorbing , i.e., for all there exists α > 0 such that for all with \(|\lambda |\geqslant \alpha \).

  4. (d)

    For all there exists such that V + V ⊆ U.

FormalPara Proof

(a) It is sufficient to show that the mapping a x is continuous. It is a consequence of Theorem 1.2 (and the definition of the product topology on E × E) that the mapping

$$\displaystyle \begin{aligned} \hspace{19pt}j_x\colon E\to E\times E,\ y\mapsto(x,y) \end{aligned}$$

is continuous. Then a x = a ∘ j x is continuous, because the addition a is continuous. The last statement is then obvious. (Note that the topology is determined if for each point in the space one knows a neighbourhood base.)

(b) Similarly to (a), we note that the mapping

is continuous. Then the continuity of m λ = m ∘ j λ follows from the continuity of the scalar multiplication m.

(c) Similarly to part (a) one shows that the mapping is continuous. Therefore there exists α > 0 such that for all with \(|\lambda |\leqslant \alpha \).

(d) Let . Then, by the continuity of the addition at the point (0, 0), there exist such that V 1 + V 2 ⊆ U. Then V := V 1 ∩ V 2 is as asserted. □

Next we introduce the concept of dual pairs of vector spaces, a central notion in our treatment.

A dual pair 〈E, F〉 consists of two vector spaces E, F over the same field and a bilinear mapping . The mapping b gives rise to mappings

  • b 1: E → F ∗, defined by b 1(x) := 〈x,  ⋅ 〉 ,

  • b 2: F → E ∗, defined by b 2(y) := 〈 ⋅ , y〉  ,

where E ∗, F ∗ denote the algebraic duals of E, F, respectively. The dual pair is separating in E if

  • , 〈x, y〉 = 0 implies that x = 0, i.e., b 1 is injective,

separating inF if

  • , 〈x, y〉 = 0 implies that y = 0, i.e., b 2 is injective,

and separating, if it is separating in E and F.

The weak topology σ(E, F) on E with respect to the dual pair 〈E, F〉 is defined as the initial topology with respect to the family ; the weak topology σ(F, E) on F is defined analogously.

If B ⊆ F is finite, then

is a σ(E, F)-neighbourhood of zero. A σ(E, F)-neighbourhood base of zero is given by

see Remark 1.6.

The following theorem is basic for the theory and important for the construction of topological vector spaces; it shows (amongst other facts) that σ(E, F) is a linear topology.

FormalPara Theorem 1.5

Let E be a vector space, a family of topological vector spaces, f ι: E → E ιlinear maps , τ the initial topology on E with respect to . Then (E, τ) is a topological vector space.

FormalPara Proof

First we show the continuity of the scalar multiplication . By Theorem 1.2 it is sufficient to show that is continuous for all . For , , one has

$$\displaystyle \begin{aligned} \hspace{19pt}f_\iota\circ m(\lambda,x)=f_\iota(\lambda x)=\lambda f_\iota(x)=m_\iota(\lambda,f_\iota(x)), \end{aligned}$$

with m ι denoting the scalar multiplication in E ι; therefore . Noting that Theorem 1.2 implies that is continuous we obtain the assertion.

The continuity of the addition a in E is proved analogously: For , the continuity of f ι ∘ a = a ι ∘ (f ι × f ι) follows from the continuity of f ι × f ι: E × E → E ι × E ι and the addition a ι in E ι. □

FormalPara Remark 1.6

If, in the situation of Theorem 1.5, is a neighbourhood base of zero, for all , then a neighbourhood base of zero for the initial topology on E is given by

This follows from (1.3) △

FormalPara Examples 1.7

(a) The weak topologies σ(E, F) and σ(F, E), for a dual pair 〈E, F〉, are linear topologies.

(b) Let E be a vector space, P a set of semi-norms on E. Then the initial topology τ P on E with respect to the mappings \( \mathop {\mathrm {id}} \nolimits \colon E\to (E,p)\) is called the topology generated by P. Theorem 1.5 implies that τ P is a linear topology.

(c) Let I be an index set. Then , with the product topology τ, the initial topology with respect to the projections , , is a topological vector space, by Theorem 1.5. With

we form the dual pair by defining the duality bracket

Then . Indeed, it is evident that , because \( \mathop {\mathrm {pr}} \nolimits _\kappa x=\langle x,\delta _\kappa \rangle \), where is defined by δ κκ := 1, δ κι := 0 if ι ≠ κ. On the other hand, for each , the mapping x↦〈x, y〉 is a finite linear combination of canonical projections, hence continuous with respect to τ.

The product topology is also generated by the family of semi-norms , where p κ(x) := |x κ| ().

(d) Let X be a topological space, E := C(X) the space of continuous functions . For compact K ⊆ X we define the semi-norm p K, by

and set

Then τ P is the topology of compact convergence ; it is a linear topology.

(e) Let be a family of topological vector spaces, and let . Then E, with the product topology, is a topological vector space. △

For a topological vector space (E, τ), the dual, or dual space, (E, τ)′ is defined as the vector space of all continuous linear functionals on E. We will not always explicitly specify the topology of a topological vector space E, and accordingly, we will denote the dual of E by E′ if it is clear from the context to which topology on E we refer.

By definition, every linear functional 〈⋅, y〉, for , is continuous for σ(E, F); the following result shows that the converse is also true.

FormalPara Theorem 1.8

Let 〈E, F〉 be a dual pair. Let . Then there exists such that η(x) = 〈x, y〉 . Expressed differently, one has (E, σ(E, F))′ = b 2(F).

For the proof we need a preparatory lemma from linear algebra.

FormalPara Lemma 1.9

Let E be a vector space, ,

$$\displaystyle \begin{aligned} \hspace{19pt}\bigcap_{j=1}^n\ker \eta_j\subseteq\ker \eta. \end{aligned}$$

Then there exist such that \(\eta =\sum _{j=1}^nc_j\eta _j\).

FormalPara Proof

(i) We start with a preliminary tool. Let F, G be vector spaces, f : E → F and g: E → G linear, g surjective, and \(\ker g\subseteq \ker f\). Then there exists \(\hat f\colon G\to F\) linear, such that \(f=\hat f\circ g\).

In fact, \(\hat f(g(x)):=f(x)\) is well-defined: If g(x) = g(x 1), then , and therefore f(x) = f(x 1). The linearity of \(\hat f\) is then easy.

(ii) Apply (i) with f = η, , to obtain . There exists a linear extension , and this extension is of the form

with suitable . Then \(\eta =\hat f\circ (\eta _1,\dots ,\eta _n)=\sum _{j=1}^nc_j\eta _j\). □

FormalPara Proof of Theorem 1.8

As η is continuous with respect to σ(E, F), there exists a finite set B ⊆ F such that

(the open unit ball in ), or expressed differently,

For with 〈x, y〉 = 0 one concludes that η(x) = 0. From Lemma 1.9 we conclude that there exist such that

□

FormalPara Example 1.10

Coming back to – see Example 1.7(c) – we note that Theorem 1.8 implies that . △

From the definition it is clear that σ(E, E′) is the coarsest linear topology on E such that E′⊇ b 2(F), and Theorem 1.8 expresses that for this topology one even has E′ = b 2(F). Later we will also obtain a finest locally convex topology with this property; see Chapter 5.

FormalPara Notes

The material of the present chapter is standard, and it is rather impossible to give precise information where the contents originated. For the fundamental notions of topology we refer to [Bou07c]; in particular, our Theorem 1.2 is as in [Bou07c, Chap. 1, § 3, Proposition 4].

Concerning topological vector spaces and in particular locally convex spaces we include at this place a list of treatises on the subject, in principle in historical order: [Ban32], [Edw65], [Köt66], [Hor66], [Sch71] (first edition 1966), [Trè67], [Gro73], [RoRo73], [Rud91], [Wil78], [Bou07a] (new edition from 1981 of [Bou64a], [Bou64b]), [Jar81] [MeVo97], [Osb14], [BoSm17]. The beginning is marked by Banach’s pioneering book. As mentioned in the preface, it was in the 1960s that the topic became “fashionable” also for teaching, and the treatises are of varying character, volume and focus. Wilansky’s contribution is notable for its richness of exercises and examples, and we add Khaleelulla’s Lecture Notes [Kha82] to the list as an abundant and well structured source of counterexamples.

The list indicated above contains only texts in which the main emphasis is on locally convex topological vector spaces. Many books on Banach space theory, functional analysis or operator theory contain also substantial parts on topological vector spaces. As examples, we mention the encyclopedic volume [DuSc58] and the treatises [Yos80], [Con90] and [Wer18].