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Functional Analysis

Functional Analysis (Analiza Functionala)

Ovidius University Constanta, 1997 (in Romanian)

 

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The preface

The Functional Analysis is a relatively recent discipline, and it is today one of the most developed branches of mathematics. Its appearance was a fundamental change of direction in mathematics, by its different approach of the problems of the Mathematical Analysis. Instead of studying particular functions and the relations and equations between them, we study these objects. Therefore we study how a differential operator or an integral transform  works not on a particular function, but on a whole class of functions, analyzing the result of the transform of these class of functions, the continuity in one ore in the other direction of the operation etc.

The present book is intended as a manual in Functional Analysis, devoted to the students of the 5th semester of the Faculty of Mathematics-Informatics of the "Ovidius" University Constanta. It is based on the classical manuals of Romulus Cristescu (Analiza functionala, editia 2, Editura didactica si pedagogica, Bucuresti, 1970, in Romanian), Nicolae Gheorghiu (Introducere in analiza  functionala,  Editura Academiei, Bucuresti, 1974, in Romanian), Gheorghe Siretchi (Analiza functionala, Universitatea din Bucuresti, Facultatea de matematica, Bucuresti, 1982, in Romanian) and Dumitru Gaspar Analiza functionala, Editura Facla, Timisoara, 1981, in Romanian). Very useful were the books of the academician Miron Nicolescu (Functii reale si elemente de topologie, Editura Didactica si Pedagogica, Bucuresti, 1958, in Romanian), as well as the book of Alexandru Ghica (Analiza functionala, Editura Academiei, Bucuresti, 1967, in Romanian), who had an important contribution to the orientation of the Romanian mathematicians towards the Functional Analysis.

Differently as in other works, taking in account the specific of our faculty, we made a parallel presentation of the Hilbert spaces and the Banach spaces from the beginning of the manual, and after that we continue with the linear topological spaces and with the important subclass of the local convex spaces. In a separate chapter, we presented the Fredholm Alternative for Hilbert spaces and for Banach spaces. The last chapter is dedicated to the distributions and the Fourier transforms, that are also important in solving problems of the technique. 

This book could be useful to all those that wish to initiate themselves in Functional Analysis: mathematicians, chemists, engineers, teachers, undergraduate students, etc.

 

Vasile Ene

 

The Contents

 

1         Introduction                    1
  1.1      Ordered sets. Zorn's Lemma 1
  1.2 Topological spaces 2
  1.3  Continuous functions. The initial topology. The final topology 5
  1.4  Separate spaces. Compact spaces 7
  1.5  Metric spaces  8
  1.6  Sequences and filters in topological spaces  10
2    Linear spaces  13
  2.1  The notion of a linear space  13
  2.2  Linear subspaces. Plane sets  14
  2.3  Linear independent sets. The Hamel basis  15
  2.4  Linear operators  16
  2.5  Linear functionals and hyperplanes  18
  2.6  Absorbant sets, equilibrated sets and absolutely convex sets  18
  2.7  The Minkovski functional  22
  2.8  The extension of linear real functionals  24
  2.9  The extension of linear complex functionals 26
3   Linear normed spaces 29
  3.1  Linear normed spaces  29
  3.2  Series in normed spaces. The Shauder basis  31
  3.3  The normed space L(X,Y)  32
  3.4  Summable families of elements of Banach spaces  36
  3.5  The completion of a linear normed space  39
  3.6  The inversion of the linear operations in normed spaces. Equivalent normed spaces  40
  3.7  Finite dimensional normed spaces  42
  3.8  Direct sums of normed spaces  44
  3.9  Approximation in normed spaces  45
  3.10  The normed linear factor space  46
  3.11  The inequalities of Hölder and Minkowski  47
  3.12  Examples of normed spaces. Banach spaces  49
  3.13  The form of the linear and continuous functionals on some normed spaces  51
  3.14  The extension of the linear and continuous functionals on normed spaces  59
  3.15  The principle of uniform boundedness  61
  3.16  The principle of the open application  62
  3.17  The principle of the closed graph  65
4    Linear topological spaces  69
  4.1  The notion of a linear topological space  69
  4.2  The properties of the linear topological spaces  70
  4.3  The characterization of the linear topological spaces  74
  4.4  Separation conditions in vectorial topological spaces  76
  4.5  Linear quasi-normed spaces  76
  4.6  Linear metrizable spaces  78
  4.7  Bounded sets in a vectorial topological space  81
  4.8  Complete vectorial topological spaces. Equivalent conditions for completeness  83
  4.9  Compact and precompact sets in vectorial topological spaces  86
  4.10  The Kolmogorov criterium of the normalization of a vectorial topological space  87
  4.11  The notion of a locally convex space  89
  4.12  The topology defined by a familiy of semi-norms  89
  4.13  Semi-norms on locally convex spaces  91
  4.14  Directed sets of semi-norms  92
  4.15  The comparation of the locally convex topologies  92
  4.16  Linear and continuous operators in vectorial topological spaces  93
  4.17  Linear and continuous operators in locally convex spaces  94
  4.18  The characterization of the convergent sequences in a locally convex space  95
  4.19  Linear and continuous functionals  95
  4.20  Locally convex metrizable spaces  97
  4.21  Factor spaces and products of vectorial topological spaces  99
  4.22  The principle of the open application. The principle of the closed graph  101
  4.23  Toneled spaces  105
  4.24 Bornological spaces   106
  4.25  Echicontinuous applications. The principle of echicontinuity  108
  4.26  Separation theorems in vectorial topological spaces  111
  4.27  Extreme points. The Krein-Milmann Theorem  113
  4.28  Finite dimensional vectorial topological spaces. The Tihonov Theorem  115
  4.29  The topological characterization of the finite dimensional vectorial topological spaces  116
  4.30  Weak topolgies on locally convex spaces  118
  4.31  Weak convergence in normed spaces  119
  4.32  Reflexivity. Weak topologies on normed spaces  121
  4.33 Inductive limits of vectorial topological spaces and locally convex spaces 125
5   Hilbert spaces 129
  5.1  Prehilbert spaces. Hilbert spaces 129
  5.2  Orthogonality  134
  5.3  The decomposition of a Hilbert spac  136
  5.4  The dual of a Hilbert space  138
  5.5  Orthonormed families  140
  5.6  Orthonormal bases  144
  5.7 Isomorphisms of Hilbert spaces 145
  5.8 Separable Hilbert spaces 147
  5.9 Applications in the space L2([a,b]) 150
  5.10 Linear operators in Hilbert spaces 154
  5.11 Adjoint operators 155
  5.12 Selfadjoint operators 158
  5.13 Normal operators 160
  5.14 Unit operators 160
  5.15 Projectors 162
  5.16 The Lax-Milgram Lemma 163
6   Normed algebras 167
  6.1  Algebras 167
  6.2   Algebras with involutions 168 
  6.3  Ideals 170 
  6.4 Normed algebras. Banach algebras 171
  6.5 Inversion properties in Banach algebra 173
  6.6 Spectral properties in Banach algebras 174
  6.7 Factor algebras 178
  6.8 Banach fields. The Ghelfan-Mazur Theorem 178
  6.9 The character space 179
  6.10 The algebra Φ(X) 180
  6.11 The Stone-Weierstrass Theorem 183
  6.12 The representation of a C*-algebra 184
  6.13 Compact operators 188
  6.14 Spectral properties of the compact operators 190
  6.15 Spectral properties of the selfadjoint operators 192
7   The Fredholm alternative 199
  7.1 The Fredholm alternative in Hilbert spaces 199
  7.2 Adjoint operations 205
  7.3 The Fredholm alternative in Banach spaces 207
8   Distributions 213
  8.1 The Schwartz space 213
  8.2 The notion of distribution 216
  8.3 Distributions that vanish on an open se 219
  8.4 Distributions with compact support 220
  8.5 Temperated distributions 221
  8.6 The derivation of the distributions with several variables 222
  8.7 The derivation of the distribution with one variable 223
  8.8 The direct product of two distributions 223
  8.9 The convolution of two distributions 224
  8.10 The Fourier transform of rapidly decreasing functions 225
  8.11 The Fourier transform of temperated distributions 229
  8.12 The Fourier transform of functions with whose square is integrable 230
    Index 233
    Bibliography 239