If $\mathcal{B}(A,E)$ is banach then $E$ is banach ($A$ is a set)

Question

Consider $\mathcal{B}(A,E)$ the space of bounded functions from $A$ to $E$ where $A$ is a set and $E$ a linear normed vector space. I need to proof that if $\mathcal{B}(A,E)$ is Banach then $E$ is Banach.

I found proofs of this in the case that $A$ is also a linear normed vector space, and those proofs involved linear functionals. But given a Cauchy sequence in $E$ how can I construct a function from the set $A$? Thanks in advance.

Answer 1

The constant functions form a closed subspace of $\mathcal{B}(A,E)$ isometric to $E$.