# On the Summability of Fourier Series by Hille E., Tamarkin J. D. By Hille E., Tamarkin J. D.

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Function bounded from below on a complete metric space (X, dist ). Given x ∈ X with (x) > inf (X ), let dist (x, y) ≤ (x) − (y) for some y = x. Then, (z) = inf (X ) for some z ∈ X . It is also equivalent to the drop theorem of Daneˇs related to the theory of normal solvability of nonlinear equations , to the flower petal theorem of Penot , and to the Caristi fixed point theorem, also called the Kirk-Caristi fixed point theorem . 9 (The Drop Theorem, Daneˇs). Given two closed nonempty sets A, B in a Banach space, with B bounded and convex and dist (A, B) > 0, there exists a point a in A such that there is no other point between a and B, that is, D(a, B) ∩ A = {a}, where D(x, B) = clco[{x} ∪ B], and where clco refers to the closure of the convex hull; this set is called a “drop” because of its geometry.

Such spaces were known as “subreflexive” spaces. The name is due to Phelps  who conjectured in this paper that every Banach space is subreflexive. In the proof of this result appears a certain convex cone in E, associated with a partial ordering, to which a transfinite argument is applied (Zorn’s lemma). c. functionals by Ekeland in the original proof of his variational principle. More precisely, for s ∈ R, consider the closed convex cone with nonempty interior, C(s) = (u, a) ∈ X × R; a + s u ≤ 0 .

Proof. 2) with S = {u} and c = infx∈X (x), if by contradiction, for all u ∈ −1 ([c, c + 2ε]) ∩ S2δ , we had 4ε (v) ≥ . δ Then, η(1, v) would be in c−ε ∩ Sδ , which is impossible since c−ε = ∅. ” When combining the quantitative deformation lemma with the compactness condition (PS)c , we obtain the useful result known widely in literature as the deformation lemma. 4 (Standard Deformation Lemma). Let c ∈ R and consider : X → R a C 1 -functional satisfying (PS)c . If c is a regular value of , then for every ε sufficiently small there exists η ∈ C([0, 1] × X ; X ) such that Comments and Additional Notes i. 