preface to the third edition
preface to the second edition
preface to the first edition
chapter1 nermed vector spaces
1.1 introduction
1.2 vector spaces
1.3 normed spaces
1.4 knach spaces
1.s linear mappings
1.6 contraction mappings and the banach fixed point theorem
1.7 exercises
chapter2 the lebesgue integral
2.1 introduction
2.2 step functions
2.3 lebesl~e intelfable functions
2.4 the absolute value of on intei fable function
2.5 series of intelqble functions so
2.6 norm in l1r
2.7 convergence almost everywhere ss
2.8 fundamentol convergence theorems
2.9 locally integmble functions
2.10 the lebesgue integral and the riemann integral
2.11 lebesgue measure on r
2.12 complex-valued lebesgue integrable functions
2.13 the spaces lpr
2.14 lebesgue integrable functions on rn
2.15 convolution
2.16 exercises
chapter3 hilbert spaces and orthonormal systems
3.1 introduction
3.2 inner product spaces
3.3 hilbert spaces
3.4 orthogonal and orthonormal systems
3.5 trigonometric fourier series
3.6 orthogonal complements and projections
3.7 linear functionals and the riesz representation theorem
3.8 exercises
chapter4 linear operators on hilbert spaces
4.1 introduction
4.2 examples of operators
4.3 bilinear functionals and quadratic forms
4.4 adjoint and seif-adjoint operators
4.5 invertible, normal, isometric, and unitary operators
4.6 positive operators
4.7 projection operators
4.8 compact operators
4.9 eigenvalues and eigenvectors
4.10 spectral decomposition
4.11 unbounded operators
4.12 exercises
chapter5 applications to integral and differential equations
5.1 introduction
5.2 basic existence theorems
5.3 fredholm integral equations
5.4 method of successive approximations
5.5 volterra integral equations
5.6 method of solution for a separable kernel
5.7 volterra integral equations of the first kind and abel''s
integral equation
5.8 ordinary differential equations and differential
operators
5.9 sturm-liouville systems
5.10 inverse differential operators and green''s functions
5.11 the fourier transform
5.12 applications of the fourier transform to ordinary
differential equations and integral equations
6.13 exercises
chapter6 generalized functions and partial differential
equations
6.1 introduction
6.2 distributions
6.3* sobolevspaces
6.4 fundamental solutions and green''s functions for partial
differential equations
6.5 weak solutions of elliptic boundary value problems
6.6 examples of applications of the fourier transform to partial
differential equations
6.7 exercises
chapter7 mathematical foundations of @uantum mechanics
7.1 introduction
7.2 basic concepts and equations of classical mechanics
poisson''s brackets in mechanics
7.3 basic concepts and postulates of quantum mechanics
7.4 the heisenberg uncertainty principle
7.5 the schrodinger equation of motion
7.6 the schrodinger picture
7.7 the heisenberg picture and the heisenberg equation of
motion
7.8 the interaction picture
7.9 the linear harmonic oscillator
7.10 angular momentum operators
7.11 the dirac relativistic wave equation
7.12 exercises
chapter8 wavelets and wavelet transforms
8.1 brief historical remarks
8.2 continuous wavelet transforms
8.3 the discrete wavelet transform
8.4 multirosolution analysis and orthonormal bases of
wavelets
8.5 examples of orthonormal wavelets
8.6 exercises
chapter9 optimization problems and other miscellaneous
applications
9.1 introduction
9.2 the gateaux and frechet differentials
9.3 optimization problems and the euler-lagrange equations
9.4 minimization of quadratic functionals s0s
9.5 variational inequalities s07
9.6 optimal control problems for dynamical systems
9.7 approximation theory
9.8 the shannon samplingtheorem
9.9 linear and nonlinear stability
9.10 bifurcation theory
9.11 exercises
hints and answers to selected exercises
bibliography
index
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