Linear functional analysis; introduction to Lebesgue integration and infinite-dimensional problems

Cover......Page __sk_0000.djvu
Copyright......Page __sk_0002.djvu
Contents......Page __sk_0007.djvu
Preface......Page __sk_0003.djvu
Notation......Page __sk_0005.djvu
1. Set-theoretic Notation......Page __sk_0011.djvu
2. Metric Spaces......Page __sk_0012.djvu
3. Elementary Topology of Metric Spaces......Page __sk_0015.djvu
4. Convergence and Completeness......Page __sk_0017.djvu
5. The Contraction Theorem......Page __sk_0020.djvu
6. Application of the Contraction Theorem to Differential Equations......Page __sk_0023.djvu
7. The Baire Category Theorem......Page __sk_0026.djvu
8. The Existence of a Continuous Nowhere-differentiable Function......Page __sk_0028.djvu
9. Continuous Functions......Page __sk_0031.djvu
10. Compactness......Page __sk_0034.djvu
1. Introductory Remarks......Page __sk_0039.djvu
2. Measure of Open Sets......Page __sk_0041.djvu
3. Measure of More General Sets......Page __sk_0046.djvu
4. Measurable Functions......Page __sk_0054.djvu
5. Integration of Non-negative Functions......Page __sk_0059.djvu
6. Integration of Real-valued and Complex-valued Functions......Page __sk_0064.djvu
7. Integration over Plane Sets......Page __sk_0067.djvu
8. Concluding Remarks......Page __sk_0070.djvu
1. Basic Concepts......Page __sk_0072.djvu
2. The Hölder and Minkowski Inequalities......Page __sk_0074.djvu
3. Definition of a Metric in L^p......Page __sk_0076.djvu
4. Completeness of L^p......Page __sk_0077.djvu
5. The Space L^∞......Page __sk_0079.djvu
6. The Spaces l_n^p and l^p......Page __sk_0080.djvu
7. Separability of L^p and l^p......Page __sk_0083.djvu
1. Linear Spaces......Page __sk_0086.djvu
2. Normed Linear Spaces......Page __sk_0090.djvu
3. Inner-product Spaces......Page __sk_0097.djvu
4. Hilbert Spaces......Page __sk_0102.djvu
5. Orthonormal Bases in Hilbert Spaces......Page __sk_0107.djvu
1. Basic Definitions and Concepts......Page __sk_0112.djvu
2. The Principle of Uniform Boundedness......Page __sk_0114.djvu
3. Bounded Linear Functionals in Hilbert Spaces......Page __sk_0119.djvu
4. The Dual Space of L^p(A)......Page __sk_0126.djvu
5. The Hahn-Banach Theorem......Page __sk_0130.djvu
1. Linear Transformations and Operators......Page __sk_0136.djvu
2. The Adjoint Operator......Page __sk_0139.djvu
3. The Inverse of an Operator......Page __sk_0141.djvu
4. Sequences of Operators......Page __sk_0145.djvu
5. Hermitian Operators......Page __sk_0147.djvu
6. Projections......Page __sk_0153.djvu
7. The Spectrum of an Operator......Page __sk_0156.djvu
8. Spectra of Hermitian, Normal, and Unitary Operators......Page __sk_0159.djvu
1. Matrix Representation of Linear Transformations......Page __sk_0163.djvu
2. Eigenvalues and Eigenvectors......Page __sk_0167.djvu
3. Finite-dimensional Inner-product Spaces......Page __sk_0172.djvu
4. Kellogg's Method of Estimating the Largest Eigenvalue......Page __sk_0178.djvu
5. Spectral Representation of Hermitian Operators......Page __sk_0181.djvu
6. Spectral Representation of Normal Operators......Page __sk_0184.djvu
1. Completely Continuous Operators......Page __sk_0187.djvu
2. Spectral Analysis of a Hermitian Completely Continuous Operator......Page __sk_0191.djvu
3. The Fredholm Alternative......Page __sk_0195.djvu
4. Survey of the Fredholm Theory of Integral Equations......Page __sk_0199.djvu
5. Estimation of the Largest Eigenvalue......Page __sk_0203.djvu
A. Partially Ordered Sets and Zorn's Lemma......Page __sk_0207.djvu
B. Concerning the Spectrum of an Operator on a Compplex Banach Space......Page __sk_0210.djvu
C. The Stieltjes Integral......Page __sk_0211.djvu
D. The Weierstrass Approximation Theorem and Approximation by Trigonometric Polynomials......Page __sk_0213.djvu
E. The Structure of Open Sets of Real Numbers......Page __sk_0217.djvu
F. Infinite Series and the Number System [0, +∞]......Page __sk_0218.djvu
G. Limit Superior and Limit Inferior......Page __sk_0221.djvu
H. The Fourier Transform in L²(R)......Page __sk_0224.djvu
Some Suggestions for Further Readings......Page __sk_0231.djvu
Cited References......Page __sk_0233.djvu
Index......Page __sk_0235.djvu