Following questions are from Linear Operators edited by Nelson Dunford and Jacob T. Schwartz, Chapter V.7, Problem 5 and Problem 7.
Let $X$ be a topological vector space (call its topology $\tau$). Given a set $\Gamma$ of linear functionals defined on $X$ (some element in $\Gamma$ could be unbounded), call $\Gamma$ total iff we have $[f(x) = 0\,\forall\,f \in \Gamma]\,\Leftrightarrow\,[x = 0]$. Use $\tau_{\gamma}$ to denote the weak topology generated by elements in $\Gamma$ and then show:
- Let $A$ be a $\tau_{\gamma}$ open subset. If there exists $r > 0$ and a $\tau$-open neighborhood of $\vec{0}$, say $V$ such that $r A \subseteq V$ (we call this $\tau$-bounded), then $X$ is finite dimensional
- If $\Gamma$ is a vector space, show that a set $S \subseteq X$ is $\tau_{\gamma}$ bounded iff $f(S)$ is bounded for all $f \in \Gamma$.
Any hints will be appreciated!
2)) $(\Rightarrow)$ Let $f\in\Gamma$ be any functional. Then $V=\{x\in X: |f(x)|<1\}$ is a $\tau_\gamma$ neighborhood of the zero of $X$. Thus there exists a positive number $M$ such that $MV\supset S$. It follows $|f(x)|\le M$ for each $x\in S$.
$(\Leftarrow)$ Let $V\in\tau_\gamma$ be any neighborhood of the zero. There exists a finite subset $F$ of $\Gamma$ such that $V\supset \{x\in X: |f(x)|\le 1\mbox{ for each }f\in F\}$. We have $M=\sup_{f\in F\, x\in S} |f(x)|<\infty$ and $S\subset MV$.