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)