Editing Infinite-dimensional holomorphy (section)

===Holomorphy===
A function ''f'' ∈ H<sub>G</sub>(U,''Y'') is '''holomorphic''' if, for every ''x'' ∈ ''U'', the [[Taylor series]] expansion
:<math>f(x+y)=\sum_{n=0}^\infty \frac{1}{n!} \widehat{D}^nf(x)(y)</math>
(which is already guaranteed to exist by Gateaux holomorphy) converges and is continuous for ''y'' in a neighborhood of 0 ∈ ''X''.  Thus holomorphy combines the notion of weak holomorphy with the convergence of the power series expansion.  The collection of holomorphic functions is denoted by H(''U'',''Y'').