Is a gateau-differentiable nonlinear functional also weakly continuous ?

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Dirk Werner

No; consider the norm (or its square) on an infinite dimensional Hilbert space.

Dirk.

Fotsa Mbogne David Jaures

Thank you for that clear counterexample.

J. W. Nieuwenhuis

Type in Google : Gateaux differentiability, and you find, for free, part of a book from Springer Verlag, about for instance Frechet and Gateaux differentiability . In this appendix you can find an example, in finite dimensions, that Gateaux differentiability does not imply continuity.

Thanks.