Question: 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.

Answer: 
**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?
Let $f\in \operatorname{Sphere}\_Y(0,1)$ then the operator
$$
T\_y: X\to Y, x\mapsto f(x) y
$$
is linear and $\Vert T\_y\Vert=\Vert y\Vert$ for any $y\in Y$.