Funktionalanalysis Teil I

Abstract

Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a topology - or more specifically, a norm - on these vector spaces, bringing both analytic and algebraic tools into play. The name functional analysis originates from the early attempts to extend calculus to functionals defined on function spaces. Results from functional analysis offer powerful tools for solving problems in the theory of (partial) differential equations, in complex analysis, and for the formulation of quantum mechanics. However, the aim of these pages is not to deal with such applications. This first part is primarily concerned with the intrinsic properties of certain classes of spaces, namely almost metric spaces, normed vector spaces and algebras, spaces of continuous and and $p$-integrable functions (for $p \in {]}0, \infty{]}$), as well as reflexive, uniformly convex, and Hilbert spaces, rather than with the study of mappings between them.