I have a few questions regarding the graph of an operator. Consider the operator $T:X \rightarrow Y$ between Banach spaces $X,Y$. Assume that $T$ is a linear operator which is (weak, weak)-continuous, so $T$ is continuous when $X$ and $Y$ are endowed with the weak topology. Consider the graph $G(T) := \{(x,y) \in X \times Y: ~~Tx = y \}$. The two questions I have are:
Can we conclude that $G(T)$ is weakly closed subspace of $X \times Y$? Also, since $G(T)$ is convex can we then use Mazur's Theorem to conclude that $G(T)$ is strongly closed in $X \times Y$?
Thanks.
The general fact, that a continuous map between Hausdorff topological spaces has closed graph implies that $G(T)$ is closed in $(X,\sigma(X,X^*))\times (Y,\sigma(Y,Y^*) = (X\times Y, \sigma(X\times Y,(X\times Y)^*))$.
You do not need Mazur's theorem to conclude that $G(T)$ is strongly closed: The strong (or norm-) topology on $X\times Y$ is finer than the weak topology and thus has more closed sets. (Mazur says the converse for convex sets.)