Tools from Functional Analysis

Abstract

Here we collect those definitions and statements which are needed in the next chapters. Section 6.1 introduces the normed spaces, Banach and Hilbert spaces as well as the operators as linear and bounded mappings between these spaces. In most of the later applications these spaces will be function spaces, containing for instance the solutions of the differential equations. It will turn out that the Sobolev spaces from Section 6.2 are well suited for the solutions of boundary value problems. The Sobolev space $H^{k}(\it\Omega)$ and $H^{k}_{0}(\it\Omega)$ for nonnegative integers k as well as $H^{s}(\it\Omega)$ for real $s\geq 0$ are introduced. The definition of the trace (restriction to the boundary Γ) will be essential in §6.2.5 for the interpretation of boundary values. To this end the Sobolev spaces $H^{s}(\it\Gamma)$ of functions on the boundary Γ must be defined (cf. Theorem 6.57). Sobolev’s Embedding Theorem 6.48 connects Sobolev spaces and classical spaces. Section 6.3 introduces dual spaces and dual mappings. Compactness properties are important for statements about the unique solvability. Compact operators and the Riesz–Schauder theory are presented in Section 6.4. The weak formulation $H^{s}(\it\Gamma)$ of the boundary-value problem is based on bilinear forms described in Section 6.5. The inf-sup condition in Lemma 6.94 is a necessary and sufficient criterion for the solvability of the weak formulation.