Problem 2
Prove that every linear functional on a finite-dimensional topological linear space is automatically continuous.
Problem 3
Let \(\mathrm{E}\) be a topological linear space. Prove that a linear functional \(f\) on \(\mathrm{E}\) is continuous if and only if a) Its null space \(\\{x f(x)=0\\}\) is closed in \(E\); b) There exists an open set \(U \subset E\) and a number \(t\) such that \(t \notin f(U)\).
Problem 8
Show that every real locally convex topological linear space E satisfying the first axiom of separation has sufficiently many continuous linear functionals. Hint. Given any nonzero element \(x_{0} \in \mathrm{E}\), show that there is a convex symmetric \(^{3}\) neighborhood \(U\) of zero such that \(x_{0} \notin U .\) Let \(p_{U}\) be the Minkowski functional of \(U\). Then, as in the proof of Theorem 6 , p. 136 , \(p_{U}\) is a finite convex functional on \(\mathrm{E}\) such that \(p_{U}(-x)=p_{U}(x)\) and $$ p_{U}(x)<1 \text { if } x \in U, \quad p_{U}\left(x_{0}\right)>1 $$ Define a linear functional \(f_{0}\left(\lambda x_{0}\right)=\mathrm{A}\) on the set \(\mathrm{L}\) of all elements of the form Ax, Clearly \(\left|f_{0}(x)\right| \leqslant p_{0}(x)\) on \(\mathrm{L}\) and \(f_{0}\left(x_{0}\right)=1\). Now use the HahnBanach theorem to extend \(f_{0}\) onto the whole space \(\mathrm{E}\). Comment. The importance of locally convex spaces is mainly due to this property (which continues to hold in the complex case).
