Linear Algebra and Its Applications, 2e + Functional Analysis Set
Wiley Series in Pure and Applied Mathematics
This set features: Linear Algebra and Its Applications, Second Edition (978-0-471-75156-4) and Functional Analysis (978-0-471-55604-6) both by Peter D. Lax.
Linear Algebra and Its Applications, Second Edition (978-0-471-75156-4)
Linear Algebra and Its Applications, Second Edition presents linear algebra as the theory and practice of linear spaces and linear maps with a unique focus on the analytical aspects as well as the numerous applications of the subject. In addition to thorough coverage of linear equations, matrices, vector spaces, game theory, and numerical analysis, the Second Edition features student-friendly additions that enhance the book's accessibility, including expanded topical coverage in the early chapters, additional exercises, and solutions to selected problems.
Beginning chapters are devoted to the abstract structure of finite dimensional vector spaces, and subsequent chapters address convexity and the duality theorem as well as describe the basics of normed linear spaces and linear maps between normed spaces.
Further updates and revisions have been included to reflect the most up-to-date coverage of the topic, including:
* The QR algorithm for finding the eigenvalues of a self-adjoint matrix
* The Householder algorithm for turning self-adjoint matrices into tridiagonal form
* The compactness of the unit ball as a criterion of finite dimensionality of a normed linear space
Additionally, eight new appendices have been added and cover topics such as: the Fast Fourier Transform; the spectral radius theorem; the Lorentz group; the compactness criterion for finite dimensionality; the characterization of commentators; proof of Liapunov's stability criterion; the construction of the Jordan Canonical form of matrices; and Carl Pearcy's elegant proof of Halmos' conjecture about the numerical range of matrices.
Clear, concise, and superbly organized, Linear Algebra and Its Applications, Second Edition serves as an excellent text for advanced undergraduate- and graduate-level courses in linear algebra. Its comprehensive treatment of the subject also makes it an ideal reference or self-study for industry professionals.
Functional Analysis (978-0-471-55604-6)
* Includes sections on the spectral resolution and spectral representation of self adjoint operators, invariant subspaces, strongly continuous one-parameter semigroups, the index of operators, the trace formula of Lidskii, the Fredholm determinant, and more.
* Assumes prior knowledge of Naive set theory, linear algebra, point set topology, basic complex variable, and real variables.
* Includes an appendix on the Riesz representation theorem.
Preface.
Preface to the First Edition.
1. Fundamentals.
2. Duality.
3. Linear Mappings.
4. Matrices.
5. Determinant and Trace.
6. Spectral Theory.
7. Euclidean Structure.
8. Spectral Theory of Self-Adjoint Mappings.
9. Calculus of Vector- and Matrix-Valued Functions.
10. Matrix Inequalities.
11. Kinematics and Dynamics.
12. Convexity.
13. The Duality Theorem.
14. Normed Linear Spaces.
15. Linear Mappings Between Normed Linear Spaces.
16. Positive Matrices.
17. How to Solve Systems of Linear Equations.
18. How to Calculate the Eigenvalues of Self-Adjoint Matrices.
19. Solutions.
Bibliography.
Appendix 1. Special Determinants.
Appendix 2. The Pfaffian.
Appendix 3. Symplectic Matrices.
Appendix 4. Tensor Product.
Appendix 5. Lattices.
Appendix 6. Fast Matrix Multiplication.
Appendix 7. Gershgorin's Theorem.
Appendix 8. The Multiplicity of Eigenvalues.
Appendix 9. The Fast Fourier Transform.
Appendix 10. The Spectral Radius.
Appendix 11. The Lorentz Group.
Appendix 12. Compactness of the Unit Ball.
Appendix 13. A Characterization of Commutators.
Appendix 14. Liapunov's Theorem.
Appendix 15. The Jordan Canonical Form.
Appendix 16. Numerical Range.
Index.
Functional Analysis
Foreword.
Linear Spaces.
Linear Maps.
The Hahn-Banach Theorem.
Applications of the Hahn-Banach Theorem.
Normed Linear Spaces.
Hilbert Space.
Applications of Hilbert Space Results.
Duals of Normed Linear Space.
Applications of Duality.
Weak Convergence.
Applications of Weak Convergence.
The Weak and Weak* Topologies.
Locally Convex Topologies and the Krein-Milman Theorem.
Examples of Convex Sets and their Extreme Points.
Bounded Linear Maps.
Examples of Bounded Linear Maps.
Banach Algebras and their Elementary Spectral Theory.
Gelfand's Theory of Commutative Banach Algebras.
Applications of Gelfand's Theory of Commutative Banach Algebras.
Examples of Operators and their Spectra.
Compact Maps.
Examples of Compact Operators.
Positive Compact Operators.
Fredholm's Theory of Integral Equations.
Invariant Subspaces.
Harmonic Analysis on a Halfline.
Index Theory.
Compact Symmetric Operators in Hilbert Space.
Examples of Compact Symmetric Operators.
Trace Class and Trace Formula.
Spectral Theory of Symmetric, Normal and Unitary Operators.
Spectral Theory of Self-Adjoint Operators.
Examples of Self-Adjoint Operators.
Semigroups of Operators.
Groups of Unitary Operators.
Examples of Strongly Continuous Semigroups.
Scattering Theory.
A Theorem of Beurling.
Appendix A: The Riesz-Kakutani Representation Theorem.
Appendix B: Theory of Distributions.
Appendix C: Zorn's Lemma.
Author Index.
Subject Index.
