Mujica's "Complex analysis in Banach spaces" exercise 1.2.B I'm trying to prove Exercise 1.2.B of Mujica's book "Complex analysis in Banach spaces" which states the following:

Let $ P: E \to F$ be a mapping such that $P|_M \in P_a(\left.^m M; F \right.)$ for each sub space $M$ of $E$ of dimension $\le m+1$. Show that $P \in P_a(\left.^m E; F \right.)$

Where $E,F$ are Banach spaces and $P_a(\left.^m M; F \right.)$ is the vector space of all the m-homogeneous polynomials from $M$ to $F$.
I don't know how to prove this. What I think is that for a given element of the space I should somehow decompose it in parts belonging to different sub spaces and then apply the hypothesis. But the hypothesis only holds for finite dimensional sub spaces and therefore I don't know how to proceed.
How should I prove this?
 A: The key word here is polarization formula. 
Precisely, if $P:E\to F$ is an $m$-homogeneous polynomial, then there exists a unique symmetric $m$-linear mapping $L_P:E\times\cdots \times E\to F$ such that $P(x)=L_P(x,\dots ,x)$, which is given by the formua
$$L_P(x_1,\dots ,x_m):=\frac{1}{2^m m!}\,\sum_{\varepsilon_1,\dots,\varepsilon_m=\pm 1} \varepsilon_1\cdots \varepsilon_m\, P\Bigl( \sum_{i=1}^m \varepsilon_i x_i\Bigr)\, .$$
Conversely, if $P:E\to F$ is such that the map $L_P$ defined by the above formula is $m$-linear, then $P$ is an $m$-homogeneous polynomial.
Now, to check the $m$-linearity of $L_P$, one never needs more than $m+1$ vectors at a time. For example, to check linearity with respect to $x_1$, you need to show (i) that $L_P(\lambda x_1,\dots ,x_m)=\lambda L_P(x_1,\dots ,x_m)$ for any fixed $m$ vectors $x_1,\dots ,x_m$ and every scalar $\lambda$; and (ii) that $L_P(u+v,x_2,\dots ,x_m)=L_P(u,x_2,\dots ,x_m)+L_P(v,x_2,\dots ,x_m)$ for any fixed $m+1$ vectors $u,v,x_2,\dots ,x_m$.
It follows immediately that $P$ is a homogeneous polynomial as soon as its restriction to any $(m+1)$-dimensional subspace is.
