Banach space exercise

Question

Suppose $X$ and $Y$ are Banach spaces and $T_{n} \in B(X,Y)$. If $T_{n}x_{n} \to 0$ in $Y$ for any choice of unit vectors $\{x_{n}\}$ in $X$, show that $\|T_{n}\| \to 0$.

Answer 1

You want to chose $x_n$ such that $$\frac{||T_n||}2\leq |T_n x_n|$$ which you can always do by definition of the norm.

As $|T_n x_n| \longrightarrow 0$ thus so does $||T_n||$.