Talk:Palais-Smale compactness condition

From Wikipedia, the free encyclopedia

[edit] 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)