Book contents
- Frontmatter
- Contents
- Preface to the Cambridge Edition
- 1 Foundations; Set Theory
- 2 General Topology
- 3 Measures
- 4 Integration
- 5 Lp Spaces; Introduction to Functional Analysis
- 6 Convex Sets and Duality of Normed Spaces
- 7 Measure, Topology, and Differentiation
- 8 Introduction to Probability Theory
- 9 Convergence of Laws and Central Limit Theorems
- 10 Conditional Expectations and Martingales
- 11 Convergence of Laws on Separable Metric Spaces
- 12 Stochastic Processes
- 13 Measurability: Borel Isomorphism and Analytic Sets
- Appendix A Axiomatic Set Theory
- Appendix B Complex Numbers, Vector Spaces, and Taylor's Theorem with Remainder
- Appendix C The Problem of Measure
- Appendix D Rearranging Sums of Nonnegative Terms
- Appendix E Pathologies of Compact Nonmetric Spaces
- Author Index
- Subject Index
- Notation Index
- References
6 - Convex Sets and Duality of Normed Spaces
Published online by Cambridge University Press: 06 July 2010
Book contents
- Frontmatter
- Contents
- Preface to the Cambridge Edition
- 1 Foundations; Set Theory
- 2 General Topology
- 3 Measures
- 4 Integration
- 5 Lp Spaces; Introduction to Functional Analysis
- 6 Convex Sets and Duality of Normed Spaces
- 7 Measure, Topology, and Differentiation
- 8 Introduction to Probability Theory
- 9 Convergence of Laws and Central Limit Theorems
- 10 Conditional Expectations and Martingales
- 11 Convergence of Laws on Separable Metric Spaces
- 12 Stochastic Processes
- 13 Measurability: Borel Isomorphism and Analytic Sets
- Appendix A Axiomatic Set Theory
- Appendix B Complex Numbers, Vector Spaces, and Taylor's Theorem with Remainder
- Appendix C The Problem of Measure
- Appendix D Rearranging Sums of Nonnegative Terms
- Appendix E Pathologies of Compact Nonmetric Spaces
- Author Index
- Subject Index
- Notation Index
- References
Summary
Functional analysis is concerned with infinite-dimensional linear spaces, such as Banach spaces and Hilbert spaces, which most often consist of functions or equivalence classes of functions. Each Banach space X has a dual space X′ defined as the set of all continuous linear functions from X into the field ℝ or ℂ.
One of the main examples of duality is for Lp spaces. Let (X, S, μ) be a measure space. Let 1 < p < ∞ and 1/p + 1/q = 1. Then it turns out that Lp and Lq are dual to each other via the linear functional f ↦ ∫ f g d μ for f in p and g in q. For p = q = 2, L2 is a Hilbert space, where it was shown previously that any continuous linear form on a Hilbert space H is given by inner product with a fixed element of H (Theorem 5.5.1).
Other than linear subspaces, some of the most natural and frequently applied subsets of a vector space S are the convex subsets C, such that for any x and y in C, and 0 < t < 1, we have tx + (1 – t)y ∈ C. These sets are treated in §§6.2 and 6.6. A function for which the region above its graph is convex is called a convex function. §6.3 deals with convex functions. Convex sets and functions are among the main subjects of modern real analysis.
- Type
- Chapter
- Information
- Real Analysis and Probability , pp. 188 - 221Publisher: Cambridge University PressPrint publication year: 2002
References
(1929). Sur les fonctionelles linéaires I, II.Studia Math. 1: 211–216, 223–239CrossRefGoogle Scholar
Banach, Stefan(1932). Théorie des Opérations Linéaires. Monografje Matematyczne I, Warsaw
and (1927). Sur le principe de la condensation de singularités.Fund. Math. 9: 50–61CrossRefGoogle Scholar
Bonnesen, Tommy, and Werner Fenchel (1934). Theorie der konvexen Körper. Springer, Berlin. Repub. Chelsea, New York, 1948CrossRef
*Brunn, Hermann (1887). Über Ovale and Eiflächen. Inauguraldissertation, Univ. München
*Brunn, Hermann(1889). Über Kurven ohne Wendepunkte. Habilitationsschrift, Univ. München
(1943). Sur la séparation des ensembles convexes dans un espace de Banach.Rev. Sci. 81: 277–278Google Scholar
Dunford, Nelsonand Jacob T. Schwartz, with the assistance of William G. Badé and Robert G. Bartle (1958). Linear Operators: Part I, General Theory. Interscience, New York
(1936). Zur Theorie der konvexen Mengen in linearen normierten Räumen.Studia Math. 6: 104–111CrossRefGoogle Scholar
, and (1956). Brunn-Minkowskischer Satz und Isoperimetrie.Math. Zeitschr. 66: 1–8CrossRefGoogle Scholar
(1922). Über Folgen linearer Operationen.Monatsh. für Math. und Physik 32: 3–88CrossRefGoogle Scholar
(1927). Über lineare Gleichungssyteme in linearen Räumen.J. für die reine und angew. Math. 157: 214–229Google Scholar
Hardy, Godfrey Harold, John Edensor Littlewood, and George Polya (1934). Inequalities. Cambridge University Press 2d ed., 1952, repr. 1967
(1923). On uniform limitedness of sets of functional operations.Bull. Amer. Math. Soc. 29: 309–315CrossRefGoogle Scholar
Hörmander, Lars (1964). Linear partial differential operators. Springer, Berlin
(1906). Sur les fonctions convexes et les inégalites entre les valeurs moyennes.Acta Math. 30: 175–193CrossRefGoogle Scholar
Kelley, John L., Isaac Namioka, and eight co-authors (1963). Linear Topological Spaces. Van Nostrand, Princeton. Repr. Springer, New York (1976)
(1934). Über die zusammenziehenden und Lipschitzschen Transformationen.Fund. Math. 22: 77–108CrossRefGoogle Scholar
(1864). Recherches sur le développement en séries trigonométriques des fonctions arbitraires et principalement de celles qui, dans un intervalle fini, admettent une infinité de maxima et de minima (in Latin).J. reine angew. Math. 63: 296–308; French transl. inActa Math. 36 (1913): 281–295CrossRefGoogle Scholar
Minkowski, Hermann (1897). Allgemeine Lehrsätze über die konvexen Polyeder. Nachr. Ges. Wiss. Göttingen Math. Phys. Kl. 1897: 198–219; Gesammelte Abhandlungen, 2 (Leipzig and Berlin, 1911, repr. Chelsea, New York, 1967), pp. 103–121
Minkowski, Hermann(1901). Sur les surfaces convexes fermées. C. R. Acad. Sci. Paris 132: 21–24; Gesammelte Abhandlungen, 2, pp. 128–130
Minkowski, Hermann(1910). Geometrie der Zahlen. Teubner; Leipzig; repr. Chelsea, New York, 1953
(1911, posth.). Theorie der konvexen Körper, insbesondere Begründung des Oberflächenbegriffs.Gesammelte Abhandlungen, 2, pp. 131–229Google Scholar
(1970). On the extension of Lipschitz, Lipschitz-Hölder continuous, and monotone functions.Bull. Amer. Math. Soc. 76: 334–339CrossRefGoogle Scholar
(1910). Untersuchungen über Systeme integrierbarer Funktionen.Math. Annalen 69: 449–497CrossRefGoogle Scholar
Roberts, Arthur Wayne, and Dale E. Varberg (1973). Convex Functions. Academic Press, New York
Rockafellar, R. Tyrrell (1970). Convex Analysis. Princeton University Press
(1930). Über die Umkehrung linearer, stetiger Funktionaloperationen.Studia Math. 2: 1–6CrossRefGoogle Scholar