Functional Analysis

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  Applied Functional Analysis. Approximation Methods and Computers

  Applied Functional Analysis, Approximation Methods and Computers

  • S.S. Kutateladze(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  Finally, Section 6 will be devoted to the use of the ideas of Functional Analysis for the construction of solution methods of a certain class of ex­ tremal problems of mathematical as well as purely industrial character. More detailed contents of each of the sections may be found in the cor­ responding introduction. 1. Some Facts From Functional Analysis The aim of the present section is, on one hand, to communicate to the reader the facts originated from classical Functional Analysis which will be necessary in the sequel; on the other hand, the exposition of certain applications of these facts to applied analysis, so that from the general theorems we shall obtain results whose direct proof is sometimes rather difficult. 1.1. Normed linear spaces and operators in them The basis of all further considerations is the notion of normed linear space.3 A normed linear space is a linear (or vector) set, i.e., a set X = {x} of elements of any nature for which operations of addition x + y and the multi­ 1Besides the questions considered here, one should recall in this connection the impor­ tant applications of Functional Analysis to mathematical physics (S. L. Sobolev) and to theoretical physics (I. M. Gelfand). 2Section 5 and 6 constitute the second part of the present work. (Unfortunately, this second part has never been written. -Ed.note.) 3See [1] as well as [2]. 172 L. V. KANTOROVICH SELECTED WORKS: PART II 173 the n-dimensional vector space with this norm will be denoted by mn. It is easy to check that here all the conditions mentioned above are met. The plication of any element by a real number Ax satisfying the usual algebraic laws are defined. Besides, for every element x, the norm ||x|| is defined: it is a real number which possesses the properties of the length of a vector. To be more precise, the norm must satisfy the following condition: 1. ||x|| > 0; ||x|| = 0 if and only if x = 0. 2. ||* + y | | < N | + ||y||.

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  Linear Operator Equations: Approximation And Regularization

  Approximation and Regularization

  • M Thamban Nair(Author)
  • 2009(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 2 Basic Results from Functional Analysis In this chapter we recall some of the basic concepts and results from Func-tional Analysis and Operator Theory. Well-known results are stated with-out proofs. These concepts and results are available in standard textbooks on Functional Analysis, for example the recent book [51] by the author. However, we do give detailed proofs of some of the results which are par-ticularly interested to us in the due course. 2.1 Spaces and Operators 2.1.1 Spaces Let X be a linear space (or vector space) over K , the field of real or complex numbers. Members of X are called vectors and members of K are called scalars . A linear space endowed with a norm is called a normed linear space . Recall that a norm on a linear space X is a non-negative real-valued func-tion x → x , x ∈ X, which satisfies the following conditions: (i) ∀ x ∈ X , x = 0 ⇐⇒ x = 0, (ii) αx = | α | x ∀ x ∈ X, ∀ α ∈ K , (iii) x + y ≤ x + y ∀ x, y ∈ X. It is easily seen that the map ( x, y ) → x -y , ( x, y ) ∈ X × X, 7 8 Linear Operator Equations: Approximation and Regularization is a metric on X . It also follows from the inequality (iii) above that x -y ≥ x - y ∀ x, y ∈ X, so that the map x → x , x ∈ X , is a uniformly continuous function from X to K with respect to the above metric on X . In due course, convergence of sequences in a normed linear space and continuity of functions between normed linear spaces will be with respect to the above referred metric induced by the norm on the spaces. NOTATION: For a convergent sequence ( s n ) in a subset of a metric space with lim n →∞ s n = s , we may simply write ‘ s n → 0’ instead of writing ‘ s n → 0 as n → ∞ ’. If a normed linear space is complete with respect to the induced metric, then it is called a Banach space .

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  Mathematics

  • A. Cemal Eringen(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  Part IV Functional Analysis Ian N. Sneddon SIMSON PROFESSOR OF MATHEMATICS UNIVERSITY OF GLASGOW, GLASGOW, SCOTLAND Introduction 356 1. Set Theory 357 1.1. Notation 357 1.2. The Algebra of Sets 358 1.3. Mappings 360 1.4. Countable Sets 365 1.5. Classes of Subsets 366 1.6. Set Functions 369 2. Vector Spaces 372 2.1. Definition of a Vector Space 373 2.2. Linear Mappings 374 2.3. The Algebraic Dual of a Vector Space 375 2.4. Convex Sets in a Vector Space 375 2.5. Maximal Subspaces and Hyperplanes 378 2.6. Linear Transformations from R n into R m in Terms of Coordinates 378 3. Topological Spaces 380 3.1. Open Sets 380 3.2. Closed Sets 383 3.3. Metric Spaces 385 3.4. Continuous Mappings 387 3.5. HausdorfT Spaces 388 3.6. Compact Sets 390 3.7. Complete Metric Spaces 393 3.8. Homeomorphisms 398 4. Topological Vector Spaces 400 4.1. Definition of a Topological Vector Space 401 4.2. Normed Vector Spaces 402 4.3. Vector Subspaces 408 4.4. Banach Spaces 409 4.5. Finite-Dimensional Normed Spaces 413 355 356 IAN N. SNEDDON 4.6. Hubert Spaces 416 4.7. Locally Convex Spaces 437 5. Spectral Theory of Linear Operators 440 5.1. The Spectrum of an Operator 440 5.2. Normal Operators in a Hubert Space 441 5.3. Self-Adjoint Operators in a Hubert Space 441 5.4. Compact Symmetric Operators in a Hubert Space 444 6. Differential Calculus 445 6.1. The Gateaux Derivative 446 6.2. The Frechet Derivative 447 6.3. The Chain Rule 448 6.4. Newton's Method 450 6.5. Higher Derivatives 451 6.6. Differentiable Manifolds 451 7. Distributions 453 7.1. The Space of Test Functions 455 7.2. Distributions 458 7.3. Examples of Distributions 459 7.4. Differentiation of Distributions 461 7.5. Multiplication of Distributions 468 7.6. Distributions with Compact Support 469 7.7. Tensor Product of Distributions 469 7.8. Convolution of Distributions 470 7.9. Fourier Transforms 472 7.10. Sobolev Spaces 475 Appendix: The Lebesgue Integral 476 A.l. Lebesgue Measure 476 A.2. The Lebesgue Integral 479 A.3.

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  Applied Functional Analysis

  • Jean-Pierre Aubin(Author)
  • 2011(Publication Date)
  • Wiley-Interscience

    (Publisher)

  - 2nd ed. p. cm. - (Pure and applied mathematics series) “A Wiley-Interscience publication.” Includes bibliographical references and index. ISBN 0471-17976-0 (alk. paper) 1. Functional Analysis. 2. Hilbert space. I. Title. 11. Series: Pure and applied mathematics (John Wiley & Sons : unnumbered) QA320.A913 1999 515’.7-d~21 99-15355 CIP 10 9 8 7 6 5 4 3 2 To my children, Anne Laure, who studied thefirst edition of this book when she was a student; Henri-Jean and Marc, who escaped this chore; and to Pierre-Cyril, who may regard in 20 years this new edition as an historical document. This page intentionally left blank CONTENTS Preface Introduction: A Guide to the Reader 1. The Projection Theorem 1.1. Definition of a Hilbert Space, 4 1.2. Review of Continuous Linear and Bilinear Operators, 10 1.3. Extension of Continuous Linear and Bilinear Operators by Density, 13 1.4. The Best Approximation Theorem, 15 1.5. Orthogonal Projectors, 18 1.6. Closed Subspaces, Quotient Spaces, and Finite Products of Hilbert Spaces, 22 * 1.7. Orthogonal Bases for a Separable Hilbert Space, 23 2. Theorems on Extension and Separation 2.1. 2.2. 2.3. 2.4. 2.5. *2.6. *2.7. *2.8. Extension of Continuous Linear and Bilinear Operators, 28 A Density Criterion, 29 Separation Theorems, 30 A Separation Theorem in Finite Dimensional Spaces, 32 Support Functions, 32 The Duality Theorem in Convex Optimization, 34 Von Neumann’s Minimax Theorem, 39 Characterization of Pareto Optima, 45 3. Dual Spaces and Transposed Operators 3.1. The Dual of a Hilbert Space, 50 3.2. Realization of the Dual of a Hilbert Space, 54 3.3. Transposition of Operators, 56 3.4. Transposition of Injective Operators, 57 xiii 1 4 27 49 vii viii CONTENTS 3.5. Duals of Finite Products, Quotient Spaces, and Closed or Dense Subspaces, 60 3.6. The Theorem of Lax-Milgram, 64 *3.7. Variational Inequalities, 65 *3.8. Noncooperative Equilibria in n-Person Quadratic Games, 67 4. The Banach Theorem and the BanachSteinhaus Theorem 4.1.

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  Aspects of Differential Geometry V

  • Esteban Calviño-Louzao, Eduardo García-Río, Peter Gilkey, JeongHyeong Park, Ramón Vázquez-Lorenzo(Authors)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  1 C H A P T E R 16 Functional Analysis We shall discuss work of the following mathematicians, among others, in Chapter 16: C. Arzelà G. Ascoli R. Baire S. Banach V. Bunyakovsky (1847-1912) (1843–1896) (1874–1932) (1892–1945) (1804–1889) A.L. Cauchy M. Fréchet E. Fredholm J. Gram D. Hilbert (1789–1857) (1878–1973) (1866–1927) (1850-1916) (1862–1943) T. Levi–Civita H. Minkowski C. Neumann J. Peetre F. Riesz (1873–1941) (1864–1909) (1832–1925) (1935–2019) (1880–1956) S. Saks J. Schauder E. Schmidt K. H. Schwarz K. Weierstrass (1897-1942) (1899–1943) (1876-1959) (1843–1921) (1815–1897) 2 16. Functional Analysis To keep this book as self-contained as possible, we present in Section 16.1 a brief review of some elementary notions in geometry and topology. In Sections 16.2 and 16.3, we establish some standard results concerning Banach and Hilbert spaces that we will need subsequently. Section 16.4 treats the spectral theory of compact self-adjoint operators in Hilbert space. Let Z D f0; ˙1; ˙2; : : : g be the ring of integers, let R denote the real numbers, let C denote the complex numbers, let F 2 denote the field with 2 elements, and let Z 2 be the Abelian group with 2 elements. 16.1 BASIC CONCEPTS IN GEOMETRY AND TOPOLOGY We begin in Section 16.1.1 by discussing the quotient topology; this is a very general con- struction that we use in Section 16.1.2 to topologize the coset space of an isometric action by a compact group on a compact metric space and in Section 19.2.1 to discuss complex projec- tive space. We turn our attention to the smooth category in Section 16.1.3 where we discuss connections and in Section 16.1.4 where we discuss some notions of Riemannian geometry in- cluding the exterior derivative d and interior derivative ı. In Section 16.1.5, we introduce sheaf cohomology; we extend the discussion of Section 8.4 of Book II from the context of simple covers to the more general setting.

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