## Hilbert Spaces with ApplicationsBuilding on the success of the two previous editions, Introduction to Hilbert Spaces with Applications, 3E, offers an overview of the basic ideas and results of Hilbert space theory and functional analysis. It acquaints students with the Lebesgue integral, and includes an enhanced presentation of results and proofs. Students and researchers will benefit from the wealth of revised examples in new, diverse applications as they apply to optimization, variational and control problems, and problems in approximation theory, nonlinear instability, and bifurcation. The text also includes a popular chapter on wavelets that has been completely updated. Students and researchers agree that this is the definitive text on Hilbert Space theory. * Updated chapter on wavelets * Improved presentation on results and proof * Revised examples and updated applications * Completely updated list of references . |

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### Contents

Normed Vector Spaces | 1 |

The Lebesgue Integral | 39 |

Hilbert Spaces and Orthonormal Systems | 93 |

Linear Operators on Hilbert Spaces | 145 |

Applications to Integral and Differential Equations | 217 |

Generalized Functions and Partial Differential Equations | 287 |

Mathematical Foundations of Quantum Mechanics | 351 |

Wavelets and Wavelet Transforms | 433 |

Optimization Problems and Other Miscellaneous Applications | 477 |

Hints and Answers to Selected Exercises | 547 |

565 | |

571 | |

### Common terms and phrases

adjoint Banach space bifurcation bilinear boundary conditions bounded operator called Cauchy sequence classical mechanics commute compact operator complete orthonormal Consider constant continuous function convergence corresponding deﬁned Deﬁnition denoted differential equations differential operator dimensional Dirac distribution eigenfunctions eigenvalues eigenvectors element energy Example exists ﬁnd ﬁnite ﬁrst ﬁxed Fourier transform Fréchet differential func function f given Green’s function Hence Hilbert space ifand implies inequality inﬁnite inner product space integrable function integral equations interval inverse Lebesgue integrable Lemma Letf linear functional linear operator matrix measurable momentum nonzero normed space obtain orthogonal orthonormal sequence orthonormal system particle polynomials positive problem Proof prove quantum mechanics satisﬁes satisfy scalars Section self-adjoint operator Show space H step functions subset subspace theory tion vector space wavelet