---
tag: 8D5Y3JNF09
kind: definition
name: Holomorphy on Banach spaces
msc: [46G20, 46T25]
defines: E8N3EC0YH5
uses: [06GZ956657, 06EH97V2A7, 0NAHXR1FRH, EAZF9NJW12, H1KHRWNH8B]
sources:
  - url: https://en.wikipedia.org/wiki/Infinite-dimensional_holomorphy
---

## Definition

Let $X$ and $Y$ be complex [[Banach spaces|06GZ956657]], let $U \subseteq X$ be open and let $f \colon U \to Y$.
The map $f$ is *holomorphic* on $U$ if, for each $a \in U$, there is a continuous complex-linear map $L \colon X \to Y$ with
$$\lim_{h \to 0} \frac{\norm{f(a + h) - f(a) - L h}}{\norm{h}} = 0.$$
That is, $f$ has a complex [[Fréchet derivative|06EH97V2A7]] $Df(a) = L$ at each point $a$ of $U$.

## Remarks

The following conditions on $f \colon U \to Y$ are equivalent:

1. $f$ is holomorphic.
2. $f$ is [[analytic|H1KHRWNH8B]] on $U$ over $\C$.
3. $f$ is continuous, and for all $a \in U$ and $h \in X$, the map $z \mapsto f(a + z h)$ is a [[holomorphic function|0NAHXR1FRH]] on the open set of $z \in \C$ with $a + z h \in U$.

A map with the property on complex lines in 3, without continuity, is called Gateaux holomorphic, after the [[Gateaux derivative|EAZF9NJW12]].

Example: let $B$ be a complex Banach algebra and $f(x) = x^2$.
Then $f(a + h) - f(a) = a h + h a + h^2$, and $\norm{h^2} \le \norm{h}^2$, so $Df(a) h = a h + h a$, which is complex-linear.

Non-example: on $\C$, the map $f(z) = \overline{z}$ has the real derivative $h \mapsto \overline{h}$, which is not complex-linear.
So $f$ is not holomorphic, although it is real Fréchet differentiable.

A discontinuous linear form on an infinite-dimensional complex Banach space is holomorphic on every complex line but is not holomorphic, since it is not continuous.
This shows that continuity in clause 3 cannot be dropped.