Examines linear structures, the topology of metric spaces, and continuity in infinite dimensions, with detailed coverage at the graduate level Includes applications to geometry and differential equations, numerous beautiful illustrations, examples, exercises, historical notes, and comprehensive index May be used in graduate seminars and courses or as a reference text by mathematicians, physicists, and engineers
Author(s): Mariano Giaquinta, Giuseppe Modica
Edition: 1
Publisher: Birkhäuser Boston
Year: 2007
Language: English
Pages: 488
Cover......Page 1
Preface......Page 5
Table of Contents......Page 9
Part I - Linear Algebra......Page 20
1.1 - The Linear Spaces R^n and C^n......Page 21
1.2 - Matrices and Linear Operators......Page 28
1.3 - Matrices and Linear Systems......Page 40
1.4 - Determinants......Page 49
1.5 - Exercises......Page 55
2.1 - Vector Spaces and Linear Maps......Page 58
2.2 - Eigenvectors and Similar Matrices......Page 74
2.2 - Exercises......Page 93
3.1 - The Geometry of Euclidean and Hermitian Spaces......Page 96
3.2 - Metrics on Real Vector Spaces......Page 112
3.3 - Exercises......Page 126
4.1 - Elements of Spectral Theory......Page 128
4.2 - Some Applications......Page 145
4.3 - Exercises......Page 160
Part II - Metrics and Topology......Page 163
5 - Metric Spaces and Continuous Functions......Page 164
5.1 - Metric Spaces......Page 166
5.2 - The Topology of Metric Spaces......Page 189
5.3 - Completeness......Page 200
5.4 - Exercises......Page 205
6.1 - Compactness......Page 211
6.2 - Extending Continuous Functions......Page 219
6.3 - Connectedness......Page 224
6.4 - Exercises......Page 230
7.1 - Curves in R^n......Page 232
7.2 - Curves in Metric Spaces......Page 254
7.3 - Exercises......Page 260
8 - Some Topics from the Topology of R^n......Page 261
8.1 - Homotopy......Page 262
8.2 - Some Results on the Topology of R^n......Page 284
8.3 - Exercises......Page 293
Part III - Continuity in Infinite-Dimensional Spaces......Page 295
9.1 - Linear Normed Spaces......Page 296
9.2 - Spaces of Bounded and Continuous Functions......Page 306
9.3 - Approximation Theorems......Page 314
9.4 - Linear Operators......Page 333
9.5 - Some General Principles for Solving Abstract Equations......Page 345
9.6 - Exercises......Page 355
10.1 - Hilbert Spaces......Page 362
10.2 - The Abstract Dirichlet's Principle and Orthogonality......Page 374
10.3 - Bilinear Forms......Page 379
10.4 - Linear Compact Operators......Page 389
10.5 - Exercises......Page 404
11.1 - Two Minimum Problems......Page 406
11.2 - A Theorem by Gelfand and Kolmogorov......Page 413
11.3 - Ordinary Differential Equations......Page 414
11.4 - Linear Integral Equations......Page 435
11.5 - Fourier's Series......Page 442
A - Mathematicians and Other Scientists......Page 465
B - Bibliographical Notes......Page 467
C - Index......Page 468
