about finite rank operator

Question

let $(X,\|.\|)$ be banach space and $T\colon (X,w)\to (X,\|.\|)$ is linear continuous operator . $(X,w)$ is a banach space with its weak topology. then $dim (rang (T)) < \infty$

Answer 1

Hint: There exists a finite set $x_i^{*}, 1\leq i \leq N$ such that $x_i^{*} (x)=0, 1\leq i \leq N$ implies $Tx=0$. [Use the fact that (by continuity) $|x_i^{*}(x)| <\epsilon_i, 1\leq i \leq N$ implies $\|Tx\|<1$ for suitable $x_i^{*}$'s and $\epsilon_i$'s].

Now show that $Tx \to (x_1^{*}(x),x_2^{*}(x),...,x_N^{*}(x))$ is an injective linear map from the range of $T$ to $\mathbb R^{N}$.