Examples for the Infinite Dimensional Morse Lemma

Examples are presented which show how to use the Morse lemma in specific infinite dimensional examples and what can go wrong if various hypotheses are dropped. One of the examples shows that the version of the Morse lemma using singularity theory can hold, yet the hypotheses of the Morse–Palais and Morse–Tromba lemmas fail. Another example shows how to obtain a concrete normal form in infinite dimensions using the splitting lemma and hypotheses related to those in the Morse–Tromba lemma. An example of Dancer is given which shows that for the validity of the Morse lemma in Hilbert space, some hypotheses on the higher order terms must be made in addition to smoothness, if the quadratic term is only weakly nondegenerate. A general conjecture along these lines is made.