Deformation theorems on non‐metrizable vector spaces and applications to critical point theory

Abstract

AbstractLet E be a Banach space and Φ : E → ℝ a 𝒞1‐functional. Let 𝒫 be a family of semi‐norms on E which separates points and generates a (possibly non‐metrizable) topology 𝒯𝒫 on E weaker than the norm topology. This is a special case of a gage space, that is, a topological space where the topology is generated by a family of semi‐metrics. We develop some critical point theory for Φ : (E, 𝒫) → ℝ. In particular, we prove deformation lemmas where the deformations are continuous with respect to 𝒯𝒫. In applications this yields a gain in compactness when Φ does not satisfy the Palais–Smale condition because one can work with the weak topology.We also prove some foundational results on gage spaces. In particular, we introduce the concept of Lipschitz continuity in this setting and prove the existence of Lipschitz continuous partitions of unity. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)