Series Preface
Preface
Linear Spaces
1.1 Linear spaces
1.2 Normed spaces
1.2.1 Convergence
1.2.2 Banach spaces
1.2.3 Completion of normed spaces
1.3 Inner product spaces
1.3.1 Hilbert spaces
1.3.2 Orthogonality
1.4 Spaces of continuously differentiable functions
1.4.1 HSlder spaces
1.5 Lp spaces
1.6 Compact setsSeries Preface
Preface
Linear Spaces
1.1 Linear spaces
1.2 Normed spaces
1.2.1 Convergence
1.2.2 Banach spaces
1.2.3 Completion of normed spaces
1.3 Inner product spaces
1.3.1 Hilbert spaces
1.3.2 Orthogonality
1.4 Spaces of continuously differentiable functions
1.4.1 HSlder spaces
1.5 Lp spaces
1.6 Compact sets
Linear Operators on Normed Spaces
2.1 Operators
2.2 Continuous linear operators
2.2.1 (V，W) as a Banach space
2.3 The geometric series theorem and its variants
2.3.1 A generalization
2.3.2 A perturbation result
2.4 Some more results on linear operators
2.4.1 An extension theorem
2.4.2 Open mapping theorem
2.4.3 Principle of uniform boundedness
2.4.4 Convergence of numerical quadratures
2.5 Linear functionals
2.5.1 An extension theorem for linear functionals
2.5.2 The Riesz representation theorem
2.6 Adjoint operators
2.7 Weak convergence and weak compactness
2.8 Compact linear operators
2.8.1 Compact integral operators on C(D)
2.8.2 Properties of compact operators
2.8.3 Integral operators on L2(a，b)
2.8.4 The Fredholm alternative theorem
2.8.5 Additional results on Fredholm integral equations
2.9 The resolvent operator
2.9.1 R(A) as a holomorphic function
Approximation Theory
3.1 Approximation of continuous functions by polynomials
3.2 Interpolation theory
3.2.1 Lagrange polynomial interpolation
3.2.2 Hermite polynomial interpolation
3.2.3 Piecewise polynomial interpolation
3.2.4 Trigonometric interpolation
3.3 Best approximation
3.3.1 Convexity，lower semicontinuity
3.3.2 Some abstract existence results
3.3.3 Existence of best approximation
3.3.4 Uniqueness of best approximation
3.4 Best approximations in inner product spaces，projection on
closed convex sets
3.5 Orthogonal polynomials
3.6 Projection operators
3.7 Uniform error bounds
3.7.1 Uniform error bounds for L2-approximations
3.7.2 L2-approximations using polynomials
3.7.3 Interpolatory projections and their convergence
Fourier Analysis and Wavelets
4.1 Fourier series
4.2 Fourier transform
4.3 Discrete Fourier transform
4.4 Haar wavelets
4.5 Multiresolution analysis
Nonlinear Equations and Their Solution by Iteration
5.1 The Banach fixed-point theorem
5.2 Applications to iterative methods
5.2.1 Nonlinear algebraic equations
5.2.2 Linear algebraic systems
5.2.3 Linear and nonlinear integral equations
5.2.4 Ordinary differential equations in Banach spaces
5.3 Differential calculus for nonlinear operators
5.3.1 Frechet and Gateaux derivatives
5.3.2 Mean value theorems
5.3.3 Partial derivatives
5.3.4 The Gateaux derivative and convex minimization
5.4 Newtons method
5.4.1 Newtons method in Banach spaces
5.4.2 Applications
5.5 Completely continuous vector fields
5.5.1 The rotation of a completely continuous vector field
5.6 Conjugate gradient method for operator equations
Finite Difference Method
6.1 Finite difference approximations
6.2 Lax equivalence theorem
6.3 More on convergence
Sobolev Spaces
7.1 Weak derivatives
7.2 Sobolev spaces
7.2.1 Sobolev spaces of integer order
7.2.2 Sobolev spaces of real order
7.2.3 Sobolev spaces over boundaries
7.3 Properties
7.3.1 Approximation by smooth functions
7.3.2 Extensions
7.3.3 Sobolev embedding theorems
7.3.4 Traces
7.3.5 Equivalent norms
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