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Talk:Palais–Smale compactness condition

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definition of "derivative" in the strong formulation

[edit]

An anon editor added:

Here denotes the Fréchet derivative of at .

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 if there exists such that
for all . Thus we can write and .

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)[reply]