Let $E$ and $F$ be Banach spaces. Let $(e_1,\ldots,e_n)$ be a basis for E and let $\xi_1,\ldots,\xi_n$ denote the corresponding coordinate functionals. Show that each $A \in L_a (^m E;F)$ can be uniquely represented as a sum
$A=\sum c_{j_1\ldots j_m} \xi_{j_1} \otimes \ldots \otimes \xi_{j_m} $
where $c_{j_1\ldots j_m} \in F$ and where the summation is taken over all $j_1,\ldots,j_m$ varying from $1$ to $n$. Conclude that $L_a (^m E;F)=L (^m E;F)$.
Notations: $\otimes$ is symbol of tensor product. For each $m\in \mathbb{N}$,$\;$ $L_a(^m E;F)$ denotes vector space of all m-linear mappings $A:E^m\rightarrow F$, whereas $L(^m E;F)$ denotes the subspace of all continuous members of $L_a(^m E;F)$ .
