# Continuity of a differential of a Banach-valued holomorphic map

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.