91Ó°ÊÓ

Let \(X\) be a Hilbert space and \(\left(x_{n}\right)\) be a sequence in \(X\). Show that if \(\left(x_{n}\right)\) converges weakly to \(x \in X\) and \(\left\|x_{n}\right\| \rightarrow\|x\|\) then \(\left\|x_{n}-x\right\| \rightarrow 0\)

Let \(X\) be a normed linear space. Show that the weak topology on \(X\) induced by \(X^{\prime}\) is the same as the topology induced by the norm on \(X\) if and only if \(X\) is finite dimensional.

Let \(X\) be a normed linear space. Show that a subspace \(X_{0}\) is closed if and only if it is closed with respect to the weak topology on \(X\) induced by \(X^{\prime}\)