Exercise 5, § 4

a) Let $E$ be an infinite dimensional normed space and $\mathscr{T}$ be its topology. Show that there exists on $E$ a normed space topology $\mathscr{T}'$ that is strictly finer than that of $\mathscr{T}$ and a normed space topology $\mathscr{T}''$ that is strictly coarser than that of $\mathscr{T}$ (define the neighbourhoods of 0 for these topologies, using a basis of $E$ put in the form $(a_{\alpha,n})$ where $\alpha$ varies in an infinite set of indices $A$ and $n$ in the set of integers $\geq 0$ and where $\|a_{\alpha,n}\| = 1$ for the given norm on $E$. b) Let $p$ be the norm defining the topology $\mathscr{T}'$. Show that, if $E$ is complete for the topology $\mathscr{T}$, then $p$ cannot be lower semi-continuous in $E$ for the topology $\mathscr{T}$ (use Baire's th. cf. III, p. 25, corollary). Deduce that the convex set $A$ defined by the relation $p(x) < 1$ does not contain any interior point for $\mathscr{T}$ even though all its points are internal. c) Deduce from b), that, if $E$ is complete for the topology $\mathscr{T}$, then there exists in $E$ convex sets which are not closed for $\mathscr{T}$, of which the intersection with every linear variety of finite dimension is closed for $\mathscr{T}$ (cf. II, p. 67, exerc. 12).

EXERCISES