Constructive functional analysis (Research notes in mathematics) 0273084186, 9780273084181
267 7 29MB
English Pages 203 [206] Year 1979
Table of contents :
1 The foundations of constructive mathematics
1.1. THE PHILOSOPHY OF CONSTRUCTIVE MATHEMATICS
1.2. THE LOGIC OF CONSTRUCTIVE MATHEMATICS
1.3. CLASSES AND COMPUTATIONS
1.4. SETS AND FUNCTIONS
1.5. INTUITIONISM
1.6. THE NUMBER SYSTEMS OF ANALYSIS
1.7. THE FAMILY F(X,F)
2 Metric spaces
2.1. CONVERGENCE AND COMPLETENESS
2.2. LOCATED SETS
2.3. CONNECTIVITY
2.4. COMPACTNESS
2.5. C(X,Y) AND ASCOLI'S THEOREM
2.6. THE CONSTRUCTION OF COMPACT SUBSETS
2.7. CONTINUOUS FUNCTIONS AND HOMEOMORPHISMS
2.8. LOCALLY COMPACT SPACES
2.9. ONE-POINT COMPACTIFICATIONS
3 Normed spaces and linear functionals
3.1. CONTINUITY OF LINEAR MAPPINGS
3.2. FINITE DIMENSIONAL LINEAR SPACES
3.3. SEPARATION AND EXTENSION THEOREMS
3.4. THE THICK NORM ON Hom(E,F)
4 The algebra C(X, F)
4.1. ALGEBRA SEMINORMS AND CHARACTERS OF C(X)
4.2. THE STONE-WEIERSTRASS THEOREM
4.3. APPLICATIONS OF THE STONE-WEIERSTRASS THEOREM
4.4. C(X) AS A BANACH SPACE
5 Integration on a locally compact space
5.1. POSITIVE LINEAR FUNCTIONALS ON C(X)
5.2. COMPLETE EXTENSION OF THE INTEGRAL
5.3. FULL SETS AND THE EQUALITY ON L1(μ)
5.4. NORM COMPLETENESS OF L1(μ)
5.5. COMPLEMENTED SETS AND INTEGRABLE SETS
5.6. THE CONSTRUCTION OF INTEGRABLE SETS
5.7. CONVERGENCE IN MEASURE; MEASURABLE FUNCTIONS
5.8. THE MONOTONE AND DOMINATED CONVERGENCE THEOREMS
5.9. INTEGRATION ON PRODUCT SPACES
6 Hilbert space and the functioned calculus
6.1. INNER PRODUCT SPACES
6.2. ORTHOGONAL PROJECTIONS
6.3. ORTHONORMAL BASES AND DIMENSION
6.4. THE STRONG OPERATOR TOPOLOGY ON Hom1(H,H)
6.5. APPROXIMATE EIGENVECTORS OF SELFADJOINT OPERATORS
6.6. THE FUNCTIONAL CALCULUS
6.7. THE GELFAND REPRESENTATION OF C*-ALGEBRAS
Epilogue On best approximation theory
1. ON BEST APPROXIMATION IN NORMED SPACES
2. CONCLUDING REMARKS
Appendix On constructive calculus
Problems
References
List of symbols
Index
1 The foundations of constructive mathematics
1.1. THE PHILOSOPHY OF CONSTRUCTIVE MATHEMATICS
1.2. THE LOGIC OF CONSTRUCTIVE MATHEMATICS
1.3. CLASSES AND COMPUTATIONS
1.4. SETS AND FUNCTIONS
1.5. INTUITIONISM
1.6. THE NUMBER SYSTEMS OF ANALYSIS
1.7. THE FAMILY F(X,F)
2 Metric spaces
2.1. CONVERGENCE AND COMPLETENESS
2.2. LOCATED SETS
2.3. CONNECTIVITY
2.4. COMPACTNESS
2.5. C(X,Y) AND ASCOLI'S THEOREM
2.6. THE CONSTRUCTION OF COMPACT SUBSETS
2.7. CONTINUOUS FUNCTIONS AND HOMEOMORPHISMS
2.8. LOCALLY COMPACT SPACES
2.9. ONE-POINT COMPACTIFICATIONS
3 Normed spaces and linear functionals
3.1. CONTINUITY OF LINEAR MAPPINGS
3.2. FINITE DIMENSIONAL LINEAR SPACES
3.3. SEPARATION AND EXTENSION THEOREMS
3.4. THE THICK NORM ON Hom(E,F)
4 The algebra C(X, F)
4.1. ALGEBRA SEMINORMS AND CHARACTERS OF C(X)
4.2. THE STONE-WEIERSTRASS THEOREM
4.3. APPLICATIONS OF THE STONE-WEIERSTRASS THEOREM
4.4. C(X) AS A BANACH SPACE
5 Integration on a locally compact space
5.1. POSITIVE LINEAR FUNCTIONALS ON C(X)
5.2. COMPLETE EXTENSION OF THE INTEGRAL
5.3. FULL SETS AND THE EQUALITY ON L1(μ)
5.4. NORM COMPLETENESS OF L1(μ)
5.5. COMPLEMENTED SETS AND INTEGRABLE SETS
5.6. THE CONSTRUCTION OF INTEGRABLE SETS
5.7. CONVERGENCE IN MEASURE; MEASURABLE FUNCTIONS
5.8. THE MONOTONE AND DOMINATED CONVERGENCE THEOREMS
5.9. INTEGRATION ON PRODUCT SPACES
6 Hilbert space and the functioned calculus
6.1. INNER PRODUCT SPACES
6.2. ORTHOGONAL PROJECTIONS
6.3. ORTHONORMAL BASES AND DIMENSION
6.4. THE STRONG OPERATOR TOPOLOGY ON Hom1(H,H)
6.5. APPROXIMATE EIGENVECTORS OF SELFADJOINT OPERATORS
6.6. THE FUNCTIONAL CALCULUS
6.7. THE GELFAND REPRESENTATION OF C*-ALGEBRAS
Epilogue On best approximation theory
1. ON BEST APPROXIMATION IN NORMED SPACES
2. CONCLUDING REMARKS
Appendix On constructive calculus
Problems
References
List of symbols
Index
- Author / Uploaded
- D. S Bridges
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