Let $X$, $Y$ be normed linear spaces and $Y$ be finite dimensional. Suppose $T \in B(X,Y)$ such that $T$ is surjective. Prove that there is some $\delta > 0$ so that $B_\delta(0_Y ) \subseteq T(B_1(0_X))$.
Any help/hints on this problem would be greatly appreciated. If you are not familiar with $ B(X,Y) $, it is the set of all bounded linear operators from $ X $ to $ Y $.
Since it is surjective, there exists a finite dimensional subspace $U$ of $X$ such that the restriction of $T$ to $U$ is an isomorphism, This implies that $T_{\mid U}(B_U(0,1))$ is open and contains $B_Y(0,\delta)$. The result follows from the fact that $T_{\mid U}(B_U(0,1))\subset T(B_X(0,1)$.