《normed vector space》.pdf

2 Normed Vector Spaces For the analysis of vector spaces, it is important to impose more structure on the space than merely the algebraic conditions in Definition 1.2.1. The purpose of this chapter is to consider norms on vector spaces and some of their properties. The key concept of a norm is presented in Section 2.1. In Section 2.2 the topological concepts treated in Section 1.4 are extended to general normed spaces. In Section 2.3 these concepts are linked with dense subsets, exemplified by Weierstrass’ theorem on approximation of continuous functions by polynomials. Section 2.4 gives a short introduction to operators on normed vector spaces, and Section 2.5 deals with expansions in normed spaces in terms of bases. 2.1 Normed vector spaces Our purpose in this section is to introduce norms on complex vector spaces. Intuitively, the norm of a vector shall measure the “size” of the vector; n thus, the norm is the analogue of the concept of length of a vector x ∈ R , considered in (1.3). O. Christensen, Functions, Spaces, and Expansions: Mathematical Tools 29 in Physics and Engineering, Applied and Numerical Harmonic Analysis, c DOI 10.1007/978-0-8176-4980-7 2, Springer Science+Business Media, LLC 2010 30 2. Normed Vector Spaces Definition 2.1.1 (Norm) Let V be a complex vector space. A norm on V is a function || · || : V → R that satisfies the following three conditions: (i) ||v || ≥ 0, ∀ v ∈ V, and ||v || = 0 ⇔ v = 0; (ii) ||αv || = |α| ||v ||, ∀ v ∈ V, α ∈ C; (iii) ||v + w || ≤ ||v || + ||w ||, ∀ v, w ∈ V. A vector space equipped with a norm is called a normed vector space. In situations where more than one vector space appears, we wi