Exercise 8, Appendix 0

Let $E$ be a real vector space and $\Gamma$ a solvable group of automorphisms of $E$. Let $p$ be a semi-norm on $E$ which is invariant under $\Gamma$ and $M$ a vector subspace of $E$, invariant under $\Gamma$. Let $u$ be a linear form on $M$, invariant under $\Gamma$ and such that $|u(x)| \leq p(x)$ for all $x \in M$. Show that there exists a linear form $v$ on $E$ which is invariant under $\Gamma$ and is such that $|v| \leq p$ and that $v$ extends $u$. (Let $K$ be the set of linear forms $v$ on $E$ which extend $u$, and such that $|v| \leq p$; then $K$ is a convex subset of $E^*$, stable under $\Gamma$ and compact for the topology induced by $\sigma(E^*, E)$. Apply the corollary of IV, p. 40.)

TABLE I. — Principal types of locally convex spaces. (N.B. — « Dual » is taken in the sense of « strong dual ».)

Finite dimensional space —> Fréchet-Montel space —> Montel space

Hilbert space —> Reflexive Banach space —> Reflexive Fréchet space —> Dual of a reflexive Fréchet space —> Reflexive space

Reflexive Banach space —> Fréchet space satisfying the first axiom of countability —> Distinguished Fréchet space —> Dual of a distinguished Fréchet space —> Dual of a semireflexive space

Banach space —> Normed space —> Metrizable space —> Boronological space —> Dual of a bornological space —> Complete space —> Quasi-complete space

Fréchet space —> Ultrabornological space —> Barrelled space —> Infra-barrelled space —> Dual of a barrelled space —> Semi-barrelled space —> Semi-reflexive semi-barrelled space —> Semi-reflexive space

Barrelled space —> Dual of a Fréchet space —> Dual of a barrelled space —> Semi-reflexive space

TABLE II. — Principal bornologies on the dual of a locally convex space.

Contained in a convex balanced and strongly compact set ↓ strongly bounded ↑ Bounded for $\sigma(E', E'')$ (1) (6) ↓ Bounded for the $\mathscr{S}$-topology ↑ Weakly bounded ↓ Equicontinuous (2) ↓ Contained in a convex balanced and weakly compact set (3) ↓ Weakly Precompact (5)

N.B. — We denote by $\mathscr{S}$ a family of bounded subsets of E such that every element is contained in a set belonging to $\mathscr{S}$. A number on the side of an arrow indicates that the corresponding implication holds if the property with the same number is satisfied.

PROPERTIES

1) Whenever E is semi-reflexive; 2) whenever E is bornological (III, p. 22, prop. 10); 3) if and only if E has the Mackey topology $\tau(E, E')$; 4) if and only if E is barrelled; 5) if and only if $E'$ is quasi-complete for $\sigma(E', E)$; 6) whenever E is semi-complete (a fortiori, quasi-complete or complete) (III, p. 27, cor. 1); 7) whenever $\mathscr{S}$ consists of sets whose closed convex balanced envelope is semi-complete (III, p. 27, th. 2).

When E is a Montel space, all the preceding bornologies are identical.