Rudiments of Functional Analysis

• Published in: A Guide to Advanced Real Analysis pp. 63 - 74
• Main Author: Folland, Gerald B
• Format: Book Chapter
• Language: English
• Published: Washington DC The Mathematical Association of America 2011; Mathematical Association of America; American Mathematical Society
• Edition: 1
• Subjects: Abstract spaces; Applied mathematics; Applied statistics; Banach space; Dimensional analysis; Dimensionality; Functional analysis; Hilbert spaces; Inner products; Linear algebra; Linear transformations; Mathematical analysis; Mathematics; Matrix theory; Metric spaces; Pure mathematics; Separable spaces; Statistical physics; Statistics; Topological spaces; Topological theorems; Topological vector spaces; Topology; Vector spaces; linear functional; open mapping theorem; bounded linear map; convergence; generated by a family of linearmaps; weak topology on a normed vector space; generated by a family of seminorms; unitary map; triangle; norm; weak; normed vector space; seminorm; orthonormal basis; orthonormal sequence; Alaoglu's theorem; uniform boundedness principle; Hilbert space; inner product; Parseval's identity; orthogonal vectors; Buniakovsky's inequality; orthogonal complement; closed graph theorem; Hahn-Banach theorem; reflexive Banach space; Banach space; sesquilinearity; absolute convergence; positive definiteness; adjoint of a bounded linear map; dual space; of an orthonormal sequence; inequality; reflexive; of a normed vector space; triangle inequality; operator norm; Schwarz inequality; weak topology; closed linear map; Hermitian form; Hermitian linear map; self-adjoint linear map; Cauchy's inequality; of uniform convergence on compact sets; weak, on a normed vector space
• ISBN: 0883853434; 9780883853436
• DOI: 10.5948/UPO9780883859155.006

Functional analysis - the meeting ground of analysis and linear algebra, mostly in an infinite-dimensional setting - is a vast subject, and this brief account does no more than scratch the surface. Our object is simply to introduce some basic concepts that are of wide utility and a few fundamental theorems: just enough to support the material in the last two chapters of this book. For those who want to learn more there are many books available; Reed and Simon [14] and Rudin [18] are among the best. Normed vector spaces and bounded linear maps Let X be a vector space over the field F, where F is either ℝ or ℂ. A seminorm on X is a function p: X → [0, ∞] such that i. p(x + y) ≤ p(x) + p(y) for all x, y ∈ X (the triangle inequality); ii. p(λx) = |λ|p(x) for all x ∈ X and λ ∈ K. A norm on X is a seminorm that satisfies the additional property iii. p(x) > 0 for all x ≠ 0. Norms are generally denoted by ∥x∥ rather than p(x). A normed vector space is a vector space X equipped with a norm ∥·∥. Every normed vector space is a metric space with the metric A Banach space is a normed vector space that is complete with respect to this metric. Every normed vector space can be completed to make a Banach space; we shall exhibit an easy way to do this later in this section.