  We prove several abstract versions of the Lojasiewicz-Simon gradient
inequality for an analytic functional on a Banach space that generalize
previous abstract versions of this inequality, weakening their hypotheses and,
in particular, the well-known infinite-dimensional version of the gradient
inequality due to Lojasiewicz proved by Simon (1983). We also prove that the
optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when
the functional is Morse-Bott, improving on similar results due to Chill (2003,
2006), Haraux and Jendoubi (2007), and Simon (1996). In our article
arXiv:1903.01953, we apply our abstract Lojasiewicz-Simon gradient inequalities
to prove a Lojasiewicz-Simon gradient inequalities for the harmonic map energy
functional using Sobolev spaces which impose minimal regularity requirements on
maps between closed, Riemannian manifolds. Those inequalities for the harmonic
map energy functional generalize those of Kwon (2002), Liu and Yang (2010),
Simon (1983, 1985), and Topping (1997). In our monograph arXiv:1510.03815, we
prove Lojasiewicz--Simon gradient inequalities for coupled Yang--Mills energy
functions using Sobolev spaces which impose minimal regularity requirements on
pairs of connections and sections. Those inequalities generalize that of the
pure Yang--Mills energy function due to the first author (Theorems 23.1 and
23.17 in arXiv:1409.1525) for base manifolds of arbitrary dimension and due to
Rade (1992) for dimensions two and three.
