Reflexivity

Abstract

We start by discussing semi-reflexivity and Montel spaces and present a number of examples of function spaces. At the end we present duality properties for reflexive spaces and Montel spaces.

We start by discussing semi-reflexivity and Montel spaces and present a number of examples of function spaces. At the end we present duality properties for reflexive spaces and Montel spaces.

We recall from Chapter 3 that a locally convex space E is called semi-reflexive if it is Hausdorff and the canonical embedding κ: E↪E″ is surjective. E is called reflexive if additionally κ is continuous, where the image space is equipped with the strong topology.

From Theorems 6.7 and 6.8 we know that (E, τ) is reflexive if and only if E is semi-reflexive and quasi-barrelled, or equivalently (because always τ ⊆ β(E, E′), by Proposition 6.4) if and only if E is semi-reflexive, and τ = β(E, E′), or equivalently (by Theorem 6.14), if and only if E is semi-reflexive and barrelled.

This is the reason why in the following we will mainly discuss semi-reflexivity .

FormalPara Theorem 8.1

Let E be a Hausdorff locally convex space. Then E is semi-reflexive if and only if every bounded set in E is weakly relatively compact.

FormalPara Proof

For the necessity we note that semi-reflexivity implies that β(E′, E) = μ(E′, E). Therefore, if A ⊆ E is bounded, then A ^∘ is a μ(E′, E)-neighbourhood of zero, and there exists a σ(E, E′)-compact barrel C ⊆ E such that A ^∘⊇ C ^∘. Then A ⊆ A ^∘∘⊆ C ^∘∘ = C.

For the sufficieny we note that the condition implies that β(E′, E) = μ(E′, E), which in turn implies that (E′, β(E′, E))′ = (E′, μ(E′, E))′ = E. □

FormalPara Remark 8.2

Note that the condition in Theorem 8.1 is a generalisation of the known criterion for the reflexivity of Banach spaces. △

A semi-Montel space is a Hausdorff locally convex space in which every bounded set is relatively compact. (This terminology reminds of Montel’s theorem from complex analysis; see Example 8.4(d).) A Montel space is a quasi-barrelled semi-Montel space.

FormalPara Corollary 8.3

If E is a semi-Montel space, then E is semi-reflexive. If E is a Montel space, then E is reflexive.

FormalPara Proof

This is obvious from Theorem 8.1. □

For use in the following example (b) we mention the notation C ₀( Ω), for the space of continuous functions ‘vanishing at ∞’, on a Hausdorff locally compact space Ω:

For a function , the support is defined by .

FormalPara Examples 8.4

(a) The space s of rapidly decreasing sequences is a Fréchet–Montel space , i.e., a Fréchet space which also is semi-Montel (hence Montel, because Fréchet spaces are barrelled). Indeed, if is a bounded sequence in s, then one can choose a subsequence converging in each coordinate, and it is easy to show that this subsequence is convergent in s. Hence s is reflexive.

(b) Let $\Omega \subseteq \mathbb R^n$ be open and bounded. Then

with norms

is a Fréchet–Montel space, therefore reflexive.

Indeed, $C_0^\infty (\Omega )$ is a Fréchet space. Also, every bounded set is relatively compact because of the Arzelà–Ascoli theorem, and therefore the space is semi-Montel.

A partial description of the dual is given as follows. If , then there exist and c⩾0 such that $|\eta (f)|\leqslant cp_m(f)$ . The mapping

is linear and isometric, and therefore the Hahn–Banach theorem implies that there exists such that $\hat \eta \circ \Phi =\eta$. The Riesz–Markov theorem (see [Rud87, Theorem 2.14]) implies that there exists a family $\big (\mu _\alpha \big )_{|\alpha |\leqslant m}$ of finite Borel measures on Ω such that

For this means that

where the derivatives of the measures should be interpreted in the sense of distributions. (Strictly speaking, the last formula would only be valid for , but the distributions can be extended by continuity to .)

(c) Let $\Omega \subseteq \mathbb R^n$ be open. Then $\mathcal E(\Omega ):=C^\infty (\Omega )$, with semi-norms

(where ∥⋅∥_K denotes the sup-norm on K) is a Fréchet–Montel space, in particular reflexive.

Let be a standard exhaustion of Ω, i.e., Ω_k is open, relatively compact in Ω_k+1 , and . Define $K_k:=\overline {\Omega _k}$ . Then any compact subset of Ω is contained in some K _k; hence the topology of $\mathcal E(\Omega )$ is generated by the set ; therefore $\mathcal E(\Omega )$ is metrisable, and also it is complete. (Note that, even though we use the standard exhaustion for the proof of the above properties, the topology does not depend on the choice of the exhaustion.)

Next we sketch why $\mathcal E(\Omega )$ is semi-Montel. As an intermediate step let , and let (f _j) be a sequence in $\mathcal E(\Omega )$, . Then the sequence (f _j) is bounded on K _k+1 and equicontinuous on K _k, and by the Arzelà–Ascoli theorem there exists a $\|\cdot \|{ }_{K_k}$-Cauchy subsequence. Now let (f _j) be a bounded sequence in $\mathcal E(\Omega )$. This means that sup_jp _K,m(f _j) < ∞ for all compact K ⊆ Ω, . Applying the previous remark and a suitable diagonal procedure one obtains a subsequence which is a p _K,m-Cauchy sequence for all compact K ⊆ Ω, , i.e., a Cauchy sequence, and therefore convergent in $\mathcal E(\Omega )$.

(d) Let be open, , with semi-norms

Then $\mathcal H(\Omega )$ is a Fréchet–Montel space, therefore reflexive.

The Montel property of $\mathcal H(\Omega )$ is just Montel’s theorem, and for completeness we recall its proof. Let $H\subseteq \mathcal H(\Omega )$ be a bounded set. Let ( Ω_n) be a standard exhaustion of Ω, and for let $K_n:=\overline {\Omega _n}$. For all one has

and there exists r _n > 0 such that . Then Cauchy’s integral formula for the derivative,

$$\displaystyle \begin{aligned} \hspace{19pt}f'(z)=\frac 1{2\pi\mathrm{i}}\int_{\partial B(z,r)}\frac{f(\zeta)}{(\zeta-z)^2}{\,\mathrm d}\zeta, \end{aligned}$$

implies that $|f'(z)|\leqslant C_{n+1}(r_n/2)^{-2}$ for all , , and this estimate shows that is equicontinuous. From the Arzelà–Ascoli theorem we conclude that $H_{K_n}$ is a relatively compact subset of C(K _n).

Now, starting with a sequence (f _k) in H we can choose a subsequence such that converges in C(K _n) for all , i.e., is convergent in C( Ω). This shows that H is relatively sequentially compact in the metric space $\mathcal H(\Omega )$, hence relatively compact.

(e) Let $\Omega \subseteq \mathbb R^n$ be open,

with semi-norms

We recall that harmonic means that $\Delta f=\sum _{j=1}^n\partial _j^2 f=0$. We will explain that then H( Ω) is a Fréchet–Montel space.

(i) Let $P:=\sum _{|\alpha |\leqslant m}a_\alpha \partial ^\alpha$ be a partial differential operator with constant coefficients. Then it is easy to see that the space

is a closed subspace of $\mathcal E(\Omega )$, therefore a Fréchet–Montel space; see Theorem 8.8(b) below. In the following we will sketch why H( Ω) = E _Δ( Ω).

(ii) We recall that harmonic functions f have the mean value property, i.e., if , r > 0 are such that B[x, r] ⊆ Ω, then

$$\displaystyle \begin{aligned} \hspace{19pt}f(x)=\frac 1{\sigma_{n-1}}\int_{S_{n-1}}f(x+r\xi)\,dS(\xi). \end{aligned}$$

We refer to [Eva98, Section 2.2.2, Theorem 2] (or any other textbook on partial differential equations) for this property.

(iii) Let be a standard exhaustion of Ω, $K_k:=\overline {\Omega _k}$, $d_k:=\operatorname {dist}(K_k,\Omega \setminus \Omega _{k+1})$, and let , ρ _k⩾0, $\mathop {\mathrm {spt}} \nolimits \rho _k\subseteq B(0,d_k)$, $\int \rho _k(x)\,dx=1$, ρ _k(x) = ρ _k(y) if |x| = |y| . Then, for , the convolution ρ _k ∗ f,

$$\displaystyle \begin{aligned} \hspace{19pt}\rho_k*f(x):=\int_{\Omega_{k+1}}\rho_k(x-y)f(y)\,dy, \end{aligned}$$

is defined for , and in fact is equal to f(x), because of the mean value property of f. Differentiating under the integral sign, one concludes that f is infinitely differentiable on Ω_k, and that

(iv) From (iii) it follows that, for , there exists a constant c _k,α such that

This shows that the topology on H( Ω) defined above is the topology induced by $\mathcal E(\Omega )$. Therefore the assertion follows from (i).

(f) The Schwartz space $\mathcal S(\mathbb R^n)$, also called the space of rapidly decreasing functions , is defined by

with norms

It is standard to show that $\mathcal S(\mathbb R^n)$ is a Fréchet space. Next we show that $\mathcal S(\mathbb R^n)$ is a Montel space.

Let , (f _k) a sequence with M :=sup_kp _m+1,m+1(f _k) < ∞. We show that then there exists a p _m,m-Cauchy subsequence. Let ε > 0; choose R > 0 such that $\frac M{1+R^2}<\varepsilon$. Then

For $|\alpha |\leqslant m$ the set is ∥⋅∥_∞-bounded and equicontinuous on B[0, R], and therefore, by the Arzelà–Ascoli theorem, there exists a subsequence $(f_{k_j})_j$ such that is ∥⋅∥_∞-convergent on B[0, R], for all $|\alpha |\leqslant m$. Repeating this argument for smaller and smaller ε and choosing suitable subsequences, we obtain a p _m,m-Cauchy subsequence.

If (f _k) is a bounded sequence in $\mathcal S(\mathbb R^n)$, then the previous procedure can be carried out for arbitrary , yielding a Cauchy sequence in $\mathcal S(\mathbb R^n)$.

We mention the remarkable fact that $\mathcal S(\mathbb R)$ is isomorphic to the space s; see [MeVo97, Example 29.5(2)]. An analogous result for $\mathcal S(\mathbb R^n)$ is presented in [ReSi80, Theorem V.13].△

After these examples we come back to some further theory.

FormalPara Theorem 8.5

Let E be a reflexive Hausdorff locally convex space. Then the space (E′, β(E′, E)) is reflexive.

FormalPara Proof

Let τ be the topology of E. By hypothesis and Theorem 6.8, (E, τ) = (E″, β(E″, E′)). Therefore E‴ = (E″, β(E″, E′))′ = (E, τ)′ = E′, with β(E‴, E″) = β(E′, E). □

FormalPara Theorem 8.6

Let E be a Montel space. Then (E′, β(E′, E)) is a Montel space.

For the proof we need a preparation. Let E be a topological vector space. We define the topology τ _c on E′ to be the topology of compact convergence, i.e., the polar topology $\tau _{\mathcal M_{\mathrm c}}$ corresponding to the collection $\mathcal M_{\mathrm c}$ of compact subsets of E.

The fact proved next is, in principle, a property of a uniformly equicontinuous set of functions on a uniform space ; topological vector spaces are special uniform spaces. In fact, part of the proof is just a generalised version of the proof of the following standard property: If B is an equicontinuous set of functions on a compact metric space A, and , ε > 0, then there exists a finite set F ⊆ A such that

FormalPara Proposition 8.7

Let E be a topological vector space, and let B ⊆ E′ be equicontinuous. Then τ _c ∩ B = σ(E′, E) ∩ B.

FormalPara Proof

The inclusion ‘⊇’ follows from τ _c ⊇ σ(E′, E). For ‘⊆’ it is sufficient to show: For and compact A, there exists a finite set F ⊆ E such that

(This property expresses that each τ _c-neighbourhood in B of y ₀ contains a suitable σ(E′, E)-neighbourhood in B of y ₀.) As B is equicontinuous, there exists a balanced such that

Due to the compactness of A, there exists a finite set F ⊆ A such that A ⊆ F + U. Now let be such that . For there exists such that , which implies that

$$\displaystyle \begin{aligned} \hspace{19pt}|\langle x,y-y_0\rangle|\leqslant|\langle x-\tilde x,y\rangle|+|\langle\tilde x,y-y_0\rangle|+ |\langle\tilde x-x,y_0\rangle|\leqslant1; \end{aligned}$$

hence . □

FormalPara Proof of Theorem 8.6

The space (E′, β(E′, E)) is reflexive, by Corollary 8.3, therefore barrelled. Let B ⊆ E′ be β(E′, E)-bounded, convex and closed. Theorem 6.8 implies that B is equicontinuous, therefore σ(E′, E)-compact (by the Alaoglu–Bourbaki theorem). Now Proposition 8.7 implies that B is τ _c-compact. Since E is a Montel space, τ _c ∩ E′ = β(E′, E), and therefore B is β(E′, E)-compact. □

FormalPara Theorem 8.8

Let E be a locally convex space, F ⊆ E a closed subspace. Then:

• (a) If E is semi-reflexive, then F is semi-reflexive.

• (b) If E is a semi-Montel space, then F is a semi-Montel space.

FormalPara Proof

(a) is a consequence of Theorem 8.1, because σ(F, F′) = σ(E, E′) ∩ F (recall Corollary 2.16).

(b) is obvious. □

FormalPara Remark 8.9

The analogue of Theorem 8.8 with ‘reflexive’ instead of ‘semi-reflexive’ or ‘Montel’ instead of ‘semi-Montel’ does not hold. There even exists a Montel space with a non-reflexive closed subspace. We refer to [Sch71, Chap. IV, Exercises 19, 20] for an example. △

FormalPara Notes

The author was not able to trace the origins of (semi-)reflexivity and the (semi-)Montel property in locally convex spaces. The examples are standard in analysis. The isomorphy of $\mathcal S(\mathbb R)$ and s, mentioned in Example 8.4(f) is due to Simon [Sim71, Theorem 1]. Theorem 8.6 can be found in [Köt66, VI, § 27.2], [Sch71, Chap. IV, § 5.9].