There are a few equivalent definitions of holomorphic functions between Banach spaces. Let $F:\mathbb{C}^n \to \ell^2$ be a function where $\ell^2$ denotes the Hilbert space of sequences $(a_1, a_2, \ldots)$ such that $\sum_{k=1}^{\infty} |a_k|^2 < \infty$. Consider the following two definitions of holomorphic:
$F = (F_1, F_2,\ldots)$ is said to be (weakly) holomorphic if for every continuous linear functional $T$ on $\ell^2$, $T \circ f$ is holomorphic on $\mathbb{C}^n$
$F = (F_1, F_2,\ldots)$ is said to be holomorphic if each $F_k$ is holomorphic and the sum $\sum_{k=1}^\infty |F_k|^2$ converges uniformly on all compact subsets of $\mathbb{C}^n.$
How do we show that the two definitions are equivalent. I can prove that definition 2 implies definition 1:
Proof that 2) implies 1):
Since $\ell^2$ is a Hilbert space with complex inner product given by $$ \langle a, b \rangle = \sum_{k=1}^{\infty} \overline{a_k}b_k, $$ $\ell^2 = (\ell^2)^*$. So it suffices to show that for every function $F$ holomorphic in the sense of definition 2 and every $a \in \ell^2$, that $$ \langle a, F\rangle(\cdot) = \sum_{k=1}^{\infty} \overline{a_k}f_k(\cdot) $$ converges uniformly on compact subsets of $\mathbb{C}^n$. This is immediate by the Cauchy-Schwarz inequality since $\left|\sum_{k=N}^{\infty} \overline{a_k}f_k(z)\right|$ is bounded above by $$ ||a||_{\ell^{2}} \sqrt{\sum_{k=N}^{\infty} |f_k|^2} $$ which can be made arbitrarily small on any compact set by taking $N$ sufficiently large.