It is a consequence of the Morse–Bott Lemma (see Theorems 2.10 and 2.14) that a \(C^2\) Morse–Bott function on an open neighborhood of a critical point in a Banach space obeys a Łojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses (Theorems 1, 2, and Corollary 3) for analytic functions on Banach spaces: If the Łojasiewicz 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 Łojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space (Łojasiewicz in Ensembles Semi-analytiques, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 1965) and the Morse Lemma (Theorems 4 and 5) 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 Łojasiewicz–Simon gradient inequalities for analytic functions on Banach spaces (Theorems 9 and 10).

莫尔斯-博特引理（见定理2.10和2.14）的结果是\（C ^ 2 \）Banach空间中临界点的开放邻域上的Morse-Bott函数服从Łojasiewicz梯度不等式，且最优指数为一半。在本文中，我们证明了Banach空间上解析函数的反函数（定理1、2和推论3）：如果解析函数的Łojasiewicz指数在临界点处等于一半，则该函数为Morse-Bott，因此它附近的关键点是一个解析子流形。我们证明中的主要成分是在有限维向量空间上的解析函数的Łojasiewicz梯度不等式（Ensembles Semi-analytiques中的Łojasiewicz，《上等科学研究院》，伊夫特河畔布尔斯，1965年）和Morse引理（定理） 4和5）用于具有退化临界点的Banach空间上的函数，这些临界点概括了文献中的先前版本，