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Linear Functional Analysis

Table of Contents

1. Preliminaries

1.1 Linear Algebra
1.2 Metric Spaces
1.3 Lebesgue Integration

2. Normed Spaces

2.1 Examples of Normed Spaces
2.2 Finite-dimensional Normed Spaces
2.3 Banach Spaces

3. Inner Product Spaces, Hilbert Spaces

3.1 Inner Products
3.2 Orthogonality
3.3 Orthogonal Complements
3.4 Orthonormal Bases in Infinite Dimensions
3.5 Fourier Series

4. Linear Operators

4.1 Continuous Linear Transformations
4.2 The Norm of a Bounded Linear Operator
4.3 The Space B(X, Y)
4.4 Inverses of Operators

5. Duality and the Hahn-Banach Theorem

5.1 Dual Spaces
5.2 Sublinear Functionals, Seminorms and the Hahn-Banach Theorem
5.3 Hahn-Banach Theorem in Normed Spaces

5.4 The General Hahn-Banach theorem
5.5 The Second Dual, Reflexive Spaces and Dual Operators
5.6 Projections and Complementary Subspaces
5.7 Weak and Weak-* Convergence

6. Linear Operators on Hilbert Spaces

6.1 The Adjoint of an Operator
6.2 Normal, Self-adjoint and Unitary Operators
6.3 The Spectrum of an Operator
6.4 Positive Operators and Projections

7. Compact Operators

7.1 Compact Operators
7.2 Spectral Theory of Compact Operators
7.3 Self-adjoint Compact Operators

8. Integral and Differential Equations

8.1 Fredholm Integral Equations
8.2 Volterra Integral Equations
8.3 Differential Equations
8.4 Eigenvalue Problems and Green's Functions

9. Solutions to Exercises

Further Reading

References

Notation Index

Index

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