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