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I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces".

We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the mappings $P:E\longrightarrow F$ such that exists a multilinear symmetric $A:E^m \longrightarrow F$ that satisfies: $$P(x) = Ax^m = A(x,\ldots,x) \quad \forall x \in E$$

Let $(P_j)$ a sequence in $\mathcal{P}_a(^mE;F)$ such that the limit $P(x)=\lim P_j(x)$ exists for every $x\in E$.

  1. Show that $P\in \mathcal{P}_a(^mE;F)$.
  2. Show that if each $P_j$ is continuous, then $P$ is continuous as well and $(P_j)$ converges uniformly to $P$ on compact subsets of $E$.

It's easy to prove (1), and the first part of item (2) is a consequence of the Principle of Uniform Boundedness to homogeneous polynomials:

Theorem: A subset of $\mathcal{P}_a(^mE;F)$ is norm bounded if and only if it is pointwise bounded.

I couldn't prove the uniform convergence over a compact subset $K\subset E$, can anyone help me? I will be grateful for any help you can provide.

OBS: This is a crosspost. Although I asked this question in math.stackexchange.com and put a bounty for 2 weeks, I still don't have a clue how to solve it. Here is the link to the original post.

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    $\begingroup$ Obviously you have uniform convergence on finite sets. So you could apply that $\sup_j \|P_j\|<\infty$ and a compact set admits a finite $\varepsilon$-net for every $\varepsilon>0$. $\endgroup$
    – M.González
    Mar 27, 2016 at 8:12

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