Here are two definitions.
ANALYTIC MAP
Let $ E$ and $ F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $ x=0$ if $$ f(x)=\sum_{k=0}^{\infty}a_k(x) $$ where
- for each $ k$, the map $a_k \colon E^k\to F $ is continuous, symmetric and $ k$-multilinear.
- $ \sum_{k=0}^{\infty}\| a_k \|\| x \|^k<\infty$.
SUMMABLE FAMILY
A family $ (u_i)_{i\in I}$ of elements of $ E$ (Banach space) is summable with sum $S$ if for every $ \epsilon>0$, there exists a finite subset $ I_0\subset I$ such that for every finite subset $ I_0\subset K\subset I$ we have
$$ \left\| \sum_{i\in K}u_i-S \right\|<\epsilon. $$
ONE EASY POINT
I can prove that the family $(a_k(x))_{k\in\mathbb{N}}$ is summable.
MY PARTICULAR CASE
I am interested in analytic maps $f \colon \mathbb{R}^n\to F $. Each $ a_k$ can then be written under the form
$$ a_k(x)=\sum_{\alpha\in \Lambda_k}a_{\alpha}x^{\alpha} $$ where $ \Lambda_k$ is the set of multiindex of size $ k$ (i.e., $\sum_i\alpha_i=k$) and $x^{\alpha}=x_1^{\alpha_1}\ldots x_n^{\alpha_n} $.
I write $ \Lambda=\bigcup_{i=0}^{\infty}\Lambda_i$ the set of all multiindex.
MY PURPOSE
My purpose is to show that I can write
$$ f(x)=\sum_{\alpha\in \Lambda} a_{\alpha}x^{\alpha}. $$
To prove that the latter sum is equal to $ \sum_{k=0}^{\infty}a_k(x)$ would imply a lot of associativity of the sum.
MY QUESTION
Is the family $ (a_{\alpha}x^{\alpha})_{\alpha\in\Lambda} $ summable?
