  It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth
Morse-Bott function on an open neighborhood of a critical point in a Banach
space obeys a Lojasiewicz gradient inequality with the optimal exponent one
half. In this article we prove converses for analytic functions on Banach
spaces: If the Lojasiewicz exponent of an analytic function is equal to one
half at a critical point, then the function is Morse-Bott and thus its critical
set nearby is an analytic submanifold. The main ingredients in our proofs are
the Lojasiewicz gradient inequality for an analytic function on a
finite-dimensional vector space and the Morse Lemma for functions on Banach
spaces with degenerate critical points that generalize previous versions in the
literature, and which we also use to give streamlined proofs of the
Lojasiewicz-Simon gradient inequalities for analytic functions on Banach
spaces.
