Re: higher order frechet derivatives

You can get this quite easily by considering the restriction of f to
finite dimensional subspaces.


marwie@xxxxxx wrote:
Hi,

I have a simple question about second order Frechet derivatives.

Consider a Banach space V and a function f:V -> R, whose second order
Frechet derivative exists at some x in V. According to the Wikipedia
article on Frechet derivatives the second derivative of f at x can be
viewed as a bilinear form on V, i.e.

D^2 f(x): VxV -> R, (y,z) -> D^2 f(x)(y,z).

If V is a finite-dimensional vector space this bilinear form is
symmetric since it is given by the Hessian matrix H_ij(x) =
(d/dx_i)(d/dx_j)f(x), which is symmetric because the partial
derivatives commute.

Is this also true on infinite-dimensional vector spaces V? And what
about second order frechet derivatives of maps between two Banach
spaces V and W?

.