Continuity of a differential of a Banach-valued holomorphic map

Question

Originally posted on MSE.

Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able to prove that the differential map $D\varphi:U\times\mathbb{C}^{n}\to F$ defined by $$D\varphi (z,v)= \lim\limits_{t\to 0}\frac{\varphi(z+tv)-\varphi(z)}{t}$$ is holomorphic.

Is there a reference for this assertion? (Or at least for continuity)

I tried to look into some sources on infinite-dimensional holomorphicity and could not find such a statement, but some of those sources are rather complicated, and so it is likely I missed it.

Answer 1

I don‘t know a reference but the result follows immediately from Hartogs‘ theorem (the fact that the image space is Banach is a bit of a red herring since a mapping into such a space is holomorphic if and only if is is weakly holomorphic).

Answer 2

You have a complete detailed treatment of the question of holomorphic functions (there called "analytic") with values in a Banach space in A. Dieudonné, Foundations of Modern Analysis, Acad. Press (1960, enlarged and corrected printing) Chapter 9, in particular 9.6.3 and 9.9.4 for Cauchy formulas for several (complex) variables Banach-valued functions. (indeed 9.9.4 is the weak form of Hartogs theorem for Banach-valued functions, see also 9.9 exercise 3 and the subsequent remark). This implies that, calling $\mathcal{H}(U;F)$ the space of holomorphic functions on $U$ with values in $F$, the operator $D\varphi$ is holomorphic.

This property extends to functions with values in any Hausdorff locally convex complete TVS $F$. Of course, this needs a definition. We can take the most natural one which is that
A function $f:\ U\to F$ is said holomorphic if for every $a\in U$, there exists an open polydisk $P\subset U$ of center $a$ in which $f(z)$ is the sum of an absolutely convergent power series in the variables $z_k-a_k$.
Considering a complete system of seminorms $p_i$ of $F$, and the Banach spaces $B_i=\widehat{F/ker(p_i)}$ we see that this is equivalent to say that, for every Banach space and continuous morphism $\ell:\ F\to B$, the composition $\ell\circ f$ is holomorphic.
In other words, this comes from the fact that any locally convex complete TVS $F$ is a dense subspace of the projective (or inverse) limit of Banach spaces.