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25992 results in Abstract analysis

Page 811 of 1300

  • 6 - Banach function spaces

  • D. J. H. Garling, St John's College, Cambridge
  • Book:
    Inequalities: A Journey into Linear Analysis
    Published online:
    06 July 2010
    Print publication:
    05 July 2007, pp 70-77

  • 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.

  • 1 - Measure and integral

  • D. J. H. Garling, St John's College, Cambridge
  • Book:
    Inequalities: A Journey into Linear Analysis
    Published online:
    06 July 2010
    Print publication:
    05 July 2007, pp 4-12

  • Summary

    Measure

    Many of the inequalities that we shall establish originally concern finite sequences and finite sums. We then extend them to infinite sequences and infinite sums, and to functions and integrals, and it is these more general results that are useful in applications.

    Although the applications can be useful in simple settings – concerning the Riemann integral of a continuous function, for example – the extensions are usually made by a limiting process. For this reason we need to work in the more general setting of measure theory, where appropriate limit theorems hold. We give a brief account of what we need to know; the details of the theory will not be needed, although it is hoped that the results that we eventually establish will encourage the reader to master them. If you are not familiar with measure theory, read through this chapter quickly, and then come back to it when you find that the need arises.

    Suppose that Ω is a set. A measure ascribes a size to some of the subsets of Ω. It turns out that we usually cannot do this in a sensible way for all the subsets of Ω, and have to restrict attention to the measurable subsets of Ω. These are the ‘good’ subsets of Ω, and include all the sets that we meet in practice. The collection of measurable sets has a rich enough structure that we can carry out countable limiting operations.

  • 7 - Rearrangements

  • D. J. H. Garling, St John's College, Cambridge
  • Book:
    Inequalities: A Journey into Linear Analysis
    Published online:
    06 July 2010
    Print publication:
    05 July 2007, pp 78-102

  • Summary

    Decreasing rearrangements

    Suppose that (E, ∥.∥E ) is a Banach function space and that f ∈ E. Then ∥f∥E = ∥|f|∥E , so that the norm of f depends only on the absolute values of f. For many important function spaces we can say more. Suppose for example that f ∈ Lp , where 1 < p < ∞. By Proposition 1.3.4, ∥f∥p = (p ∫ t p−1µ(|f| > t)dt)1/p , and so ∥f∥p depends only on the distribution of |f|. The same is true for functions in Orlicz spaces. In this chapter, we shall consider properties of functions and spaces of functions with this property.

    In order to avoid some technical difficulties which have little real interest, we shall restrict our attention to two cases:

  • (i) (Ω, Σ, µ) is an atom-free measure space;

  • (ii) Ω = N or {1, …, n}, with counting measure.

    • In the second case, we are concerned with sequences, and the arguments are usually, but not always, easier. We shall begin by considering case (i) in detail, and shall then describe what happens in case (ii), giving details only when different arguments are needed.

    Suppose that we are in the first case, so that (Ω, Σ, µ) is atom-free. We shall then make use of various properties of the measure space, which follow from the fact that if A ∈ Σ and 0 < t < µ(A) then there exists a subset B of A with µ(B) = t (Exercise 7.1).

  • 4 - Convexity, and Jensen's inequality

  • D. J. H. Garling, St John's College, Cambridge
  • Book:
    Inequalities: A Journey into Linear Analysis
    Published online:
    06 July 2010
    Print publication:
    05 July 2007, pp 24-44

  • Summary

    Convex sets and convex functions

    Many important inequalities depend upon convexity. In this chapter, we shall establish Jensen's inequality, the most fundamental of these inequalities, in various forms.

    A subset C of a real or complex vector space E is convex if whenever x and y are in C and 0 ≤ θ ≤ 1 then (1 − θ)x + θy ∈ C. This says that the real line segment [x, y] is contained in C. Convexity is a real property: in the complex case, we are restricting attention to the underlying real space. Convexity is an affine property, but we shall restrict our attention to vector spaces rather than to affine spaces.

    Proposition 4.1.1A subset C of a vector space E is convex if and only if whenever x 1, …, xn ∈ C and p 1, …, pn are positive numbers with p 1 + … + pn = 1 then p 1 x 1 + … + pnxn ∈ C.

    Proof The condition is certainly sufficient. We prove necessity by induction on n. The result is trivially true when n = 1, and is true for n = 2, as this reduces to the definition of convexity. Suppose that the result is true for n − 1, and that x 1, …, xn and p 1, …, pn are as above.

  • Introduction

  • D. J. H. Garling, St John's College, Cambridge
  • Book:
    Inequalities: A Journey into Linear Analysis
    Published online:
    06 July 2010
    Print publication:
    05 July 2007, pp 1-3

  • Summary

    Inequalities lie at the heart of a great deal of mathematics. G.H. Hardy reported Harald Bohr as saying ‘all analysts spend half their time hunting through the literature for inequalities which they want to use but cannot prove’. Inequalities provide control, to enable results to be proved. They also impose constraints; for example, Gromov's theorem on the symplectic embedding of a sphere in a cylinder establishes an inequality that says that the radius of the cylinder cannot be too small. Similar inequalities occur elsewhere, for example in theoretical physics, where the uncertainty principle (which is an inequality) and Bell's inequality impose constraints, and, more classically, in thermodynamics, where the second law provides a fundamental inequality concerning entropy.

    Thus there are very many important inequalities. This book is not intended to be a compendium of these; instead, it provides an introduction to a selection of inequalities, not including any of those mentioned above. The inequalities that we consider have a common theme; they relate to problems in real analysis, and more particularly to problems in linear analysis. Incidentally, they include many of the inequalities considered in the fascinating and ground-breaking book Inequalities, by Hardy, Littlewood and Pólya, originally published in 1934.

    The first intention of this book, then, is to establish fundamental inequalities in this area. But more importantly, its purpose is to put them in context, and to show how useful they are.

  • Colouring Semirandom Graphs

  • AMIN COJA-OGHLAN
  • Journal:
    Combinatorics, Probability and Computing / Volume 16 / Issue 4 / July 2007
    Published online by Cambridge University Press:
    01 July 2007, pp. 515-552

  • We study semirandom k-colourable graphs made up as follows. Partition the vertex set V = {1, . . ., n} randomly into k classes V1, . . ., Vk of equal size and include each Vi–Vj-edge with probability p independently (1 ≤ i < j ≤ k) to obtain a graph G0. Then, an adversary may add further Vi–Vj-edges (i≠j) to G0, thereby completing the semirandom graph G = G*n,p,k. We show that if np ≥ max{(1 + ϵ)klnn, C0k2} for a certain constant C0>0 and an arbitrarily small but constant ϵ>0, an optimal colouring of G*n,p,k can be found in polynomial time with high probability. Furthermore, if np ≥ C0max{klnn, k2}, a k-colouring of G*n,p,k can be computed in polynomial expected time. Moreover, an optimal colouring of G*n,p,k can be computed in expected polynomial time if k ≤ ln1/3n and np ≥ C0klnn. By contrast, it is NP-hard to k-colour G*n,p,k With high probability if .

Page 811 of 1300