In this and the following two chapters we discuss some surprising properties concerning the weak topology of Banach spaces. (However, the discussion will not be restricted to Banach spaces!)

For the first result stated below we will give an interesting and motivating application in the subsequent example. The proof of this result and the more genreral Krein–Šmulian theorem requires the consideration of several additional topologies on locally convex spaces.

FormalPara Theorem 12.1 (Banach)

Let E be a Banach space, F ⊆ E′ a subspace. Then F is σ(E′, E)-closed if and only if \(F\cap B_{E'}\)is σ(E′, E)-closed.

FormalPara Example 12.2

Let E be a complex Banach space, open, f :  Ω → E. A ‘traditional’ result is then Dunford’s theorem: f is holomorphic if and only if x′∘ f is holomorphic for all ([Dun38, Theorem 76], [Yos80, Section V.3]). (‘Holomorphic’ is defined as complex differentiable, and the -valued theory of functions of one complex variable carries over to E-valued functions, with the result that E-valued holomorphic functions are analytic.) It is relatively standard that the hypothesis in Dunford’s theorem can be weakened to the requirement that x′∘ f is holomorphic for all , where F is an almost norming subspace of E′. Using Theorem 12.1 one can show that even this condition can be replaced by a weaker requirement:

Let f :  Ω → E be locally bounded, and assume that the set

is separating in E. Then f is holomorphic.

We start the proof by noting that F is a subspace of E′, and that the hypothesis implies that F is σ(E′, E)-dense in E′; see Corollary 2.10. Now we show that the closed unit ball of F,

is σ(E′, E)-closed. We introduce the mapping , x′↦x′∘ f and note that φ is continuous with respect to σ(E′, E) and the product topology on . By Montel’s theorem – see Example 8.4(d) –, the set

is a compact subset of C( Ω) (provided with the topology of compact convergence); therefore H is closed in . Then the equality

$$\displaystyle \begin{aligned}\hspace{19pt}B_F=B_{E'}\cap\varphi^{-1}(H) \end{aligned}$$

shows that B F is σ(E′, E)-closed.

Now we conclude from Theorem 12.1 that F is σ(E′, E)-closed, and therefore F = E′. Then the assertion follows from Dunford’s theorem.

The result quoted above is due to Grosse-Erdmann ([GrE92]). The above elegant proof is a variant of the proof given by Arendt and Nikolski ([ArNi00, Theorem 3.1]); see also [ABHN11, Theorem A.7].△

For another application of Theorem 12.1, resulting in a generalisation of Pettis’ theorem on measurability of Banach space-valued functions we refer to [ABHN11, Corollary 1.3.3].

FormalPara Remark 12.3

Corollary 9.18 can be derived from Theorem 12.1. Indeed, if is σ(E′, E)-continuous on \(B_{E'}\), then \(u^{-1}(0)\cap B_{E'}\) is σ(E′, E)-closed; hence u −1(0) is a σ(E′, E)-closed subspace of E′, and u is σ(E′, E)-continuous, i.e., . △

The proof of Theorem 12.1 will be given at the end of this chapter; the remainder of the chapter is devoted to preparations for the proof of a more general version.

For a locally convex space E we define a topology τ f on E′ by

it is not difficult te check that τ f is indeed a topology on E′. Expressed differently, we equip the sets (the collection of equicontinuous subsets of E′) with the trace σ(E′, E) ∩ B of the weak topology and equip E′ with the finest topology on E′ for which all injections j B: B↪E′ are continuous. If \(\mathcal E_0\) is a cobase of \(\mathcal E\), for instance where is a neighbourhood base of zero in E, then τ f is also the finest topology for which all j B, for , are continuous. (Concerning terminology: A cobase of a collection \(\mathcal A\) of sets is a subcollection \(\mathcal A'\) of \(\mathcal A\) such that for all there exists such that A ⊆ A′.)

Clearly, a set A ⊆ E′ is τ f-closed if and only if A ∩ B is σ(E′, E) ∩ B-closed for all B belonging to a cobase \(\mathcal E_0\) of \(\mathcal E\).

FormalPara Proposition 12.4

Let E be a locally convex space, and let τ fbe the topology on E′ defined above. Then τ f ⊇ τ c (topology of compact convergence, see Chapter 8 ). The topology τ fis Hausdorff, translation invariant, and every τ f-neighbourhood of zero is absorbing and contains a balanced τ f-neighbourhood of zero.

FormalPara Proof

It was shown in Proposition 8.7 that τ c ∩ B = σ(E′, E) ∩ B for all equicontinuous sets B ⊆ E′. As τ f is the finest topology coinciding with σ(E′, E) on the equicontinuous sets, it follows that τ f ⊇ τ c.

The topology τ f is Hausdorff because τ f ⊇ σ(E′, E), and τ f is translation invariant because the collection of equicontinuous sets and the topology σ(E′, E) are translation invariant.

Let V  be a τ f-neighbourhood of zero, , B ⊆ E′ equicontinuous, balanced and containing x′. Then there exists a balanced σ(E′, E)-neighbourhood of zero W such that W ∩ B ⊆ V ∩ B. There exists such that for \(|\lambda |\leqslant \alpha \), and therefore

This shows that V is absorbing.

Let U be a τ f-neighbourhood of zero, and let

be its ‘balanced core’ (the largest balanced subset of U). Let B ⊆ E′ be equicontinuous and balanced. There exists a balanced such that W ∩ B ⊆ U ∩ B ⊆ U. Since W ∩ B is balanced, one concludes that W ∩ B ⊆ V , and this implies that W ∩ B ⊆ V ∩ B. This shows that V is a τ f-neighbourhood of zero. □

FormalPara Remark 12.5

Why can one only show ‘balanced’ in Proposition 12.4(b)? The reason in the proof is that there does not exist an ‘absolutely convex core’ of sets. In fact, the reason is deeper, because it is known that in general τ f is not a linear (let alone a locally convex) topology ([Kōm64, § 2]).

The index ‘f’ in τ f is historical and probably just stands for ‘finest’. △

FormalPara Theorem 12.6 (Banach–Dieudonné)

Let E be a metrisable locally convex space. Let

Then \(\tau_{\mathrm f}=\tau_{\mathrm c}=\tau_{\mathrm{ns}}:=\tau_{\mathcal M_{\mathrm{ns}}}\).

FormalPara Proof

(i) ‘τ f ⊇ τ c ⊇ τ ns’. The first inclusion is part of Proposition 12.4; the second inclusion holds because every is compact.

(ii) ‘τ ns ⊇ τ f’. Let U be an open τ f-neighbourhood of zero. It suffices to show that there exists such that A ∘⊆ U.

Because E is metrisable, there exists a decreasing neighbourhood base of zero in E, V 0 = E, and all V n absolutely convex and closed. In part (iii) of the proof we will show:

(∗)

Assuming this, we set \(A:=\big (\bigcup _{k=0}^\infty B_k\big )\cup \{0\}\). Then obviously . Also \(A^{\textstyle \circ }\subseteq A_n^{\textstyle \circ }\), and therefore \(A^{\textstyle \circ }\cap V_n^{\textstyle \circ }\subseteq U\) . From one obtains , and therefore A ∘⊆ U.

(iii) We prove (∗) by induction. For n = 0, the assertion holds with \(B_0=\varnothing \). Assume that B k has been obtained for k = 0, …, n − 1. We have to find a finite set B n ⊆ V n such that \((A_n\cup B_n)^{\textstyle \circ }\cap V_{n+1}^{\textstyle \circ }\subseteq U\).

Set \(C:=V_{n+1}^{\textstyle \circ }\setminus U\). The polar \(V_{n+1}^{\textstyle \circ }\) is compact for σ(E′, E), by the Alaoglu–Bourbaki theorem. Because \(V_{n+1}^{\textstyle \circ }\) is equicontinuous, the topologies τ f and σ(E′, E) agree on \(V_{n+1}^{\textstyle \circ }\); therefore, \(V_{n+1}^{\textstyle \circ }\) is also compact for τ f, and as a consequence the closed subset C is compact for τ f. Since \(A_n^{\textstyle \circ }\cap V_n^{\textstyle \circ }\subseteq U\) and \(U\cap C=\varnothing \), we know that \(A_n^{\textstyle \circ }\cap V_n^{\textstyle \circ }\cap C=\varnothing \). For all the set \(\{x\}^{\textstyle \circ }\cap A_n^{\textstyle \circ }\cap C\) is a closed subset of C, and

Now the compactness of C implies that the family cannot have the finite intersection property. This means that there exists a finite subset B n ⊆ V n such that \(\varnothing = B_n^{\textstyle \circ }\cap A_n^{\textstyle \circ }\cap C =(A_n\cup B_n)^{\textstyle \circ }\cap \big (V_{n+1}^{\textstyle \circ }\setminus U\big )\), hence \((A_n\cup B_n)^{\textstyle \circ }\cap V_{n+1}^{\textstyle \circ }\subseteq U\). □

FormalPara Remark 12.7

The usual way to formulate Theorem 12.6 is to use the topology τ pc, the topology of uniform convergence on the precompact sets of E, instead of τ c. An inspection of the proof of Proposition 8.7 immediately yields that it also shows that τ pc ∩ B = σ(E′, E) ∩ B for all equicontinuous sets B. This implies that in Theorem 12.6 one also obtains τ f = τ pc = τ ns (which is the traditional assertion in the Banach–Dieudonné theorem). △

Let E be a locally convex space,

Then \(\tau _{\mathrm {cc}}:=\tau _{\mathcal M_{\mathrm {cc}}}\), the topology of compact convex convergence, is a polar topology on E′. Observe that, in view of Lemma 4.9, the set

is a cobase of \(\mathcal M_{\mathrm {cc}}\), hence \(\tau _{\mathcal M_{\mathrm {cc}}^{\prime }}=\tau _{\mathcal M_{\mathrm {cc}}}\). Note that σ(E′, E) ⊆ τ cc ⊆ μ(E′, E); therefore (E′, τ cc)′ = b 1(E) (= E if E is Hausdorff).

If E is Hausdorff and quasi-complete, then τ c = τ cc is compatible with the dual pair 〈E, E′〉.

FormalPara Theorem 12.8 (Krein–Šmulian)

Let E be a Fréchet space, and let be a neighbourhood base of zero in E. Then a convex set A ⊆ E′ is σ(E′, E)-closed if and only if A ∩ U ∘is σ(E′, E)-closed for every .

FormalPara Proof

The necessity is trivial.

For the sufficiency, we recall that A is τ f-closed, which by Theorem 12.6 implies that A is τ c-closed. By the above preliminary remark, τ c = τ cc is compatible with the dual pair 〈E, E′〉, and therefore the convex set A is σ(E′, E)-closed as well. □

FormalPara Proof of Theorem 12.1

This follows immediately from Theorem 12.8. □

FormalPara Notes

Theorem 12.1 is contained in [Ban32, Chap. VIII, § 3, Lemme 3]. In order to understand this it should be mentioned that the subspaces of E′ whose intersection with the closed unit ball is σ(E′, E)-closed occur in [Ban32] as ‘transfiniment fermé’, whereas σ(E′, E)-closed subspaces are ‘régulièrement fermé’. A translation into more modern terminology was given by Bourbaki [Bou38], and a new proof was given by Dieudonné [Die42, Théorème 23]. (Interestingly enough, the proof by contraposition in [Ban32, Chap. VIII, § 3, Lemme 2] seems to have persisted in the literature, where usually in step (iii) of the proof of Theorem 12.6, the existence of a finite set B n is shown by contraposition.) The new methods introduced by Dieudonné then served to extend Theorem 12.8 – proved in [KrŠm40, Theorem 5] only for the case of Banach spaces – to more general settings. For this and a variety of related results obtained by these methods we refer to Köthe [Köt66, § 21.10] and Schaefer [Sch71, Chap. IV, § 6.4].