Q:

A subspace of $\mathcal{B}(X;Y)$ isometric to $Y$

Let $X$ and $Y$ be normed spaces. If $\mathcal{B}(X,Y)$ is the space of all continuous linear functions of $^{X}Y$, I'm trying to prove that $\mathcal{B}(X,Y)$ has a closed subspace isometric to $Y$. Can someone give me only a clue.

A:

Hint: On a basis $e_n$ of $X$, $f\in \mathcal B (X,Y)$ is basically (roughly) the same as a specification of $f_n\equiv f(e_n)\in Y$ for all $n$. That is, there is one value in $Y$ for each $n$. You want only one value in $Y$. What set of functions could you consider?

