Showing that $T_W \in B(W,Y)$ cannot be extended to $T \in B(X,Y)$ If $W \subset X$ and $Y=W$

2017-12-13 13:58:14

Let $X$ and $Y$ be normed spaces and let $W \subset X$ be a sublinear space. Suppose that $T_W \in B(W,Y)$, then $T_W$ can be extended to an element $T$ in $B(X,Y)$.

If we have $Y=\mathbb{F}$ then it is true even we have $\|T_W\|=\|T\|$. However this is false in general.

One of the counterexamples is if $Y=W$ and we choose $T_W$ be the identity map on $W$.

How to show this counterexample works for showing the statement above is false?