I'm trying to solve Problem 1.2.J in Mujica's "Complex Analysis in Banach Spaces". The problem states as follows:
Let $E$ and $F$ be Banach spaces over $\mathbb{K}$, with $E$ finite dimensional. Let $(e_1, \dots, e_n)$ be a basis for $E$ and let $\xi_1, \dots, \xi_n$ denote the corresponding coordinate functionals. Show that each $P \in P(\left.^m E; F \right.)$ can be uniquely represented as a sum
$$P=\sum c_{\alpha} \xi_1^{\alpha_1} \dots \xi_n^{\alpha_n}$$
where the summation is taken over all multi-indices $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n$ such that $|\alpha|=m$.
My definition of a polynomial is that $P$ is a (continuous) m-homogeneous polynomial if there is a linear (continuous) map from $E^m$ to $F$ such that $P(x)=A(x, \dots, x)$.
My approach was to try to prove this using induction. When $m=1$ the situation is clear:
$$P(x)=T(x)=\sum x_i T(e_i)$$
And this matches the given formula considering:
$$T=P=\sum T(e_i) \xi_i$$
Then for $m$ I thought of fixing a variable and to have an (m-1)-homogeneous polynomial to apply the induction hypothesis. I considered an arbitrary m-homogeneous polynomial $P_0$ and then:
$$P_0(x)=\sum x_j A(\sum x_i e_i, \dots , e_j)$$
Each one of the terms is an (m-1)-homogeneous polynomial so I can apply the hypothesis getting:
$$P_0(x)=\sum x_j \left( \sum c_{\alpha} \xi_1^{\alpha_1} \dots \xi_n^{\alpha_n} \right)$$
Where $|\alpha|=m-1$. But I can't conclude from here.
Is there any flaw in my argument? How can I prove it?