I need help with proving the following properties. My lecturer suggests using the Hahn-Banach extension theorem but I have no idea where should I begin with.
Let E, F be 2 normed spaces and $f_n$be a sequence of function in the dual space L(E;F). Suppose $f_n$ converges pointwisely to some function $f$ then $f$ is also in L(E;F) and $||f||\leq\lim||f_n||$.
Please correct me if I understood wrongly the notation L(E;F).
For every $x \in E$ the sequence $\|f_n(x)\|_F$ is bounded because $f_n(x) \to f(x)$ in $F$. The Uniform Boundedness Principle (which depends on the Baire Category Theorem, not the Hahn-Banach Theorem) implies that $$\sup_n\|f_n\| < \infty.$$ The fact that $f$ is linear should be clear. If $x \in E$ then $$\|f(x)\|_F \le \|f_n(x) - f(x)\|_F + \|f_n(x)\|_F \le \|f_n(x) - f(x)\|_F + \|f_n\|\|x\|_E$$ so that (on letting $n \to \infty$) $$\|f(x)\|_F \le \liminf_n \|f_n\| \|x\|_E.$$ Consequently $$\|f\| = \sup_{\|x\|_E \le 1} \|f(x)\|_F \le \liminf_n \|f_n\|.$$