Holomorphy on Banach spaces

Remarks

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

1. $f$ is holomorphic.
2. $f$ is analytic 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 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.

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.