## Functional Analysis, Sobolev Spaces and Partial Differential EquationsUniquely, this book presents a coherent, concise and unified way of combining elements from two distinct “worlds,” functional analysis (FA) and partial differential equations (PDEs), and is intended for students who have a good background in real analysis. This text presents a smooth transition from FA to PDEs by analyzing in great detail the simple case of one-dimensional PDEs (i.e., ODEs), a more manageable approach for the beginner. Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Moreover, the wealth of exercises and additional material presented, leads the reader to the frontier of research. This book has its roots in a celebrated course taught by the author for many years and is a completely revised, updated, and expanded English edition of the important “Analyse Fonctionnelle” (1983). Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English version is a welcome addition to this list. The first part of the text deals with abstract results in FA and operator theory. The second part is concerned with the study of spaces of functions (of one or more real variables) having specific differentiability properties, e.g., the celebrated Sobolev spaces, which lie at the heart of the modern theory of PDEs. The Sobolev spaces occur in a wide range of questions, both in pure and applied mathematics, appearing in linear and nonlinear PDEs which arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, physics etc. and belong in the toolbox of any graduate student studying analysis. |

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### Contents

Chapter 1 | 1 |

Chapter 2 | 31 |

Chapter 3 | 55 |

Chapter 4 | 88 |

Chapter 5 | 131 |

Chapter 6 | 157 |

Chapter 7 | 181 |

Chapter 8 | 200 |

Chapter 10 | 324 |

Chapter 11 | 349 |

Solutions of Some Exercises | 371 |

Problems | 435 |

Partial Solutions of the Problems | 521 |

Notation | 583 |

585 | |

595 | |

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apply Theorem Assume Banach space bijective bilinear form bounded linear bounded operator Chapter Check choose class C1 closed subspace codim compact operator conclude contradiction converges convex function convex set Corollary Deduce defined denoted dense dual space eigenvalues equation example Exercise exists a constant exists a sequence exists a unique finite finite-dimensional fixed following properties function u e Given Hahn-Banach Hilbert space Hint hyperplane inequality injection integer isometry Lemma Let u e linear operator linear subspace LP(Q nonempty norm obtain open set problem proof of Theorem Proposition Prove reflexive Remark satisfies scalar product self-adjoint Show Sobolev Spaces space and let subset surjective topology a(E vector space Vm e H weak solution weak topology weakly Wl'p write