  This paper considers optimization of smooth nonconvex functionals in smooth
infinite dimensional spaces. A H\"older gradient descent algorithm is first
proposed for finding approximate first-order points of regularized polynomial
functionals. This method is then applied to analyze the evaluation complexity
of an adaptive regularization method which searches for approximate first-order
points of functionals with $\beta$-H\"older continuous derivatives. It is shown
that finding an $\epsilon$-approximate first-order point requires at most
$O(\epsilon^{-\frac{p+\beta}{p+\beta-1}})$ evaluations of the functional and
its first $p$ derivatives.
