Lokale Darstellungen differenzierbarer Funktionen auf Banachräumen. (Local representations of differentiable functions on Banach spaces) (Q1108545)

Summary: Local topological properties of differentiable functions on Banach spaces are studied. It will be shown that in the neighborhood of a nondegenerate critical point p a certain \(C^ 2\)-function f can be written in the following from: \[ f(x)=\alpha (\lambda)+f(p)-\sum^{q}_{i=1}\alpha^ 2_ i+\sum^{m}_{i=q+1} \alpha^ 2_ i \] where m is the dimension of some m-dimensional subspace, the real numbers \(\alpha_ 1,...,a_ m\) are determined by a basis of this subspace and \(\alpha(\lambda)\) is given by a function \(\lambda\). The dimension m can tend to \(\infty\), while the number q does not change.