Editing Infinite-dimensional holomorphy (section)

====Examples====
*If ''f'' ∈ ''U'', then ''f'' has [[Gateaux derivative]]s of all orders, since for ''x'' ∈ ''U'' and ''h''<sub>1</sub>, ..., ''h<sub>k</sub>'' ∈ ''X'', the ''k''-th order Gateaux derivative ''D<sup>k</sup>f''(''x''){''h''<sub>1</sub>, ..., ''h<sub>k</sub>''} involves only iterated directional derivatives in the span of the ''h<sub>i</sub>'', which is a finite-dimensional space.  In this case, the iterated Gateaux derivatives are multilinear in the ''h<sub>i</sub>'', but will in general fail to be continuous when regarded over the whole space ''X''.
*Furthermore, a version of Taylor's theorem holds:
::<math>f(x+y)=\sum_{n=0}^\infty \frac{1}{n!} \widehat{D}^nf(x)(y)</math> 
:Here, <math>\widehat{D}^nf(x)(y)</math> is the [[homogeneous polynomial]] of degree ''n'' in ''y'' associated with the [[multilinear operator]] ''D<sup>n</sup>f''(''x''). The convergence of this series is not uniform.  More precisely, if ''V'' ⊂ ''X'' is a ''fixed'' finite-dimensional subspace, then the series converges uniformly on sufficiently small compact neighborhoods of 0 ∈ ''Y''.  However, if the subspace ''V'' is permitted to vary, then the convergence fails: it will in general fail to be uniform with respect to this variation.  Note that this is in sharp contrast with the finite dimensional case.

*[[Hartog's theorem]] holds for Gateaux holomorphic functions in the following sense:
<blockquote>If ''f'' : (''U'' ⊂ ''X''<sub>1</sub>) × (''V'' ⊂ ''X''<sub>2</sub>) → ''Y'' is a function which is ''separately'' Gateaux holomorphic in each of its arguments, then ''f'' is Gateaux holomorphic on the product space.</blockquote>