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# Banach and Hilbert Spaces

• David Gilbarg
• Neil S. Trudinger
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 224)

## Abstract

This chapter supplies the functional analytic material required for our study of existence of solutions of linear elliptic equations in Chapters 6 and 8. This material will be familiar to a reader already versed in basic functional analysis but we shall assume some acquaintance with elementary linear algebra and the theory of metric spaces. Unless otherwise indicated, all linear spaces used in this book are assumed to be defined over the real number field. The theory of this chapter, however, carries over almost unchanged if the real numbers are replaced by the complex numbers. Let 𝒱 be a linear space over ℝ. A norm on V is a mapping p : V → ℝ (henceforth we write p(x) = ‖ x ‖ = ‖ x V , xV) satisfying
1. (i)

x ‖ ⩾ 0 for all xV, ‖x‖ = 0 if and only if x = 0;

2. (ii)

‖ α x‖ = |α| ‖x‖ for all α ∊ ℝ, xV ;

3. (iii)

x + y‖ ⩽ ‖x‖ + ‖ y ‖ for all x, yV(triangle inequality).

## Keywords

Hilbert Space Banach Space Null Space Closed Subspace Normed Linear Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1977

## Authors and Affiliations

• David Gilbarg
• 1
• Neil S. Trudinger
• 2
1. 1.Department of MathematicsStanford UniversityStanfordUSA
2. 2.Department of Pure MathematicsAustralian National UniversityCanberraAustralia