Talk:Palais–Smale compactness condition

definition of "derivative" in the strong formulation

An anon editor added:

Here $I'[x]$ denotes the Fréchet derivative of $I$ at $x\in H$.

Well, yes and no. I is a functional on a Hilbert space; the domain is reflexive and the range is the reals (also a Hilbert space, but also the space of scalars). We can make a stronger statement:

I is differentiable at $u\in H$ if there exists $v\in H$ such that

$I[w]=I[u]+(v,w-u)+o(\Vert w-u\Vert )$

for all $w\in H$. Thus we can write $I':H\rightarrow H$ and $I'[u]=v$.

Dunno how to work that into the article in a sensible way. I don't think it appears anywhere else in Wikipedia; for instance, it doesn't appear in the Derivative (generalizations) article. Lunch (talk) 23:26, 27 January 2008 (UTC)