Exercise 9, § 1

Topological Vector Spaces, III, p. 38

Let E be the vector space over $\mathbf{R}$ consisting of all regulated functions on the interval $I = [0, 1]$ (FVR, I, p. 4). For every integer $n > 0$ , let $V_n$ be the set of all functions $f \in E$ such that $\int_0^1 \sqrt{|f(t)|} \, dt \leq 1/n$ . Show that the sets $V_n$ form a fundamental system of neighbourhoods of 0 for a metrizable topology which is compatible with the vector space structure of E and that for this topology the sets $V_n$ are bounded; but the convex envelope of each $V_n$ is the entire space E. (Observe that every function $f \in E$ can be written as $f = \frac{1}{2}(g + h)$ , where $g$ and $h$ belong to E, and

\int_0^1 \sqrt{|g(t)|}\ dt = \int_0^1 \sqrt{|h(t)|}\ dt = \frac{1}{\sqrt{2}} \int_0^1 \sqrt{|f(t)|}\ dt.)

Deduce from this that the only locally convex topology coarser than the topology of E is the coarsest topology on E.

Solution

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