I am new in functional analysis. I am facing a doubt about the following:
If $X$ is a finite dimensional Banach space and $Y$ is a normed linear space, then can we say that for a norm one bijective operator $T\in L(X, Y)$, $T(B(X))=B(Y)$? If not, then in what condition we can say that $T(B(X))=B(Y)$? What will happen when $X=Y$ (i.e, $T\in L(X, X)$)?
Here, $B(X)=\{x\in X:\|x\|\leq 1\}$.
I was thinking about the norm of $T^{-1}$. We know that $\|T^{-1}\|\ne \|T\|^{-1}$ in general but if the above is true then in those case we can conclude $\|T^{-1}\|= \|T\|^{-1}$. Now, I do not able to find out when $T(B(X))=B(Y)$ will hold. Please help me.