Book contents
- Frontmatter
- Contents
- Introduction
- 1 Measure and integral
- 2 The Cauchy–Schwarz inequality
- 3 The AM–GM inequality
- 4 Convexity, and Jensen's inequality
- 5 The Lp spaces
- 6 Banach function spaces
- 7 Rearrangements
- 8 Maximal inequalities
- 9 Complex interpolation
- 10 Real interpolation
- 11 The Hilbert transform, and Hilbert's inequalities
- 12 Khintchine's inequality
- 13 Hypercontractive and logarithmic Sobolev inequalities
- 14 Hadamard's inequality
- 15 Hilbert space operator inequalities
- 16 Summing operators
- 17 Approximation numbers and eigenvalues
- 18 Grothendieck's inequality, type and cotype
- References
- Index of inequalities
- Index
6 - Banach function spaces
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Introduction
- 1 Measure and integral
- 2 The Cauchy–Schwarz inequality
- 3 The AM–GM inequality
- 4 Convexity, and Jensen's inequality
- 5 The Lp spaces
- 6 Banach function spaces
- 7 Rearrangements
- 8 Maximal inequalities
- 9 Complex interpolation
- 10 Real interpolation
- 11 The Hilbert transform, and Hilbert's inequalities
- 12 Khintchine's inequality
- 13 Hypercontractive and logarithmic Sobolev inequalities
- 14 Hadamard's inequality
- 15 Hilbert space operator inequalities
- 16 Summing operators
- 17 Approximation numbers and eigenvalues
- 18 Grothendieck's inequality, type and cotype
- References
- Index of inequalities
- Index
Summary
Banach function spaces
In this chapter, we introduce the idea of a Banach function space; this provides a general setting for most of the spaces of functions that we consider. As an example, we introduce the class of Orlicz spaces, which includes the Lp spaces for 1 < p < ∞. As always, let (Ω, Σ, µ) be a σ-finite measure space, and let M = M(Ω, Σ, µ) be the space of (equivalence classes of) measurable functions on Ω.
A function norm on M is a function ρ : M → [0, ∞] (note that ∞ is allowed) satisfying the following properties:
(i) ρ(f) = 0 if and only if f = 0; ρ(αf) = |α|ρ(f) for α ≠ 0; ρ(f + g) ≤ ρ(f) + ρ(g).
(ii) If |f| ≤ |g| then ρ(f) ≤ ρ(g).
(iii) If 0 ≤ fn ↗ f then ρ(f) = lim n→∞ ρ(fn).
(iv) If A ∈ Σ and µ(A) < ∞ then ρ(IA) < ∞.
(v) If A ∈ Σ and µ(A) < ∞ there exists CA such that ∫ A|f| dµ ≤ CA ρ(f) for any f ∈ M.
If ρ is a function norm, the space E = {f ∈ M: ρ(f) < ∞} is called a Banach function space. If f ∈ E, we write ∥f∥E for ρ(f). Then condition (i) ensures that E is a vector space and that ∥.∥E is a norm on it.
- Type
- Chapter
- Information
- Inequalities: A Journey into Linear Analysis , pp. 70 - 77Publisher: Cambridge University PressPrint publication year: 2007