Banach space and weak convergence

Question

Let $T ∈\mathcal{L}(X,Y)$ (continuous mapping from $X$ to $Y$), where $X$ and $Y$ are infinite dimensional Banach spaces. For each sequence $(x_n)_n≥1$ in $X$ and for each $x ∈ X$, show that $x_n → x $ convergence weakly, as $n →∞$, implies $Tx_n → Tx$ convergence weakly, as $n →∞$.

Answer 1

Suppose that $y^{\ast}\in Y^{\ast}$, then $y^{\ast}\circ T\in X^{\ast}$. But $y^{\ast}\circ T$ is continuous as norm-to-scalar, then it must be continuous as weak-to-scalar, so $y^{\ast}\circ T(x_{n})\rightarrow y^{\ast}\circ T(x)$, that is, $y^{\ast}(T(x_{n}))\rightarrow y^{\ast}(T(x))$.

Note that $\|y^{\ast}\circ T\|\leq\|y^{\ast}\|\|T\|<\infty$ since $T$ is norm-to-norm continuous.