## Applied Functional AnalysisA novel, practical introduction to functional analysis In the twenty years since the first edition of Applied Functional Analysis was published, there has been an explosion in the number of books on functional analysis. Yet none of these offers the unique perspective of this new edition. Jean-Pierre Aubin updates his popular reference on functional analysis with new insights and recent discoveries-adding three new chapters on set-valued analysis and convex analysis, viability kernels and capture basins, and first-order partial differential equations. He presents, for the first time at an introductory level, the extension of differential calculus in the framework of both the theory of distributions and set-valued analysis, and discusses their application for studying boundary-value problems for elliptic and parabolic partial differential equations and for systems of first-order partial differential equations. To keep the presentation concise and accessible, Jean-Pierre Aubin introduces functional analysis through the simple Hilbertian structure. He seamlessly blends pure mathematics with applied areas that illustrate the theory, incorporating a broad range of examples from numerical analysis, systems theory, calculus of variations, control and optimization theory, convex and nonsmooth analysis, and more. Finally, a summary of the essential theorems as well as exercises reinforcing key concepts are provided. Applied Functional Analysis, Second Edition is an excellent and timely resource for both pure and applied mathematicians. |

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### Contents

1 | |

4 | |

2 Theorems on Extension and Separation | 27 |

3 Dual Spaces and Transposed Operators | 49 |

4 The Banach Theorem and the BanachSteinhaus Theorem | 70 |

5 Construction of Hilbert Spaces | 94 |

6 L2 Spaces and Convolution Operators | 120 |

7 Sobolev Spaces of Functions of One Variable | 145 |

11 Elementary Spectral Theory | 259 |

12 HilbertSchmidt Operators and Tensor Products | 283 |

13 Boundary Value Problems | 309 |

14 DifferentialOperational Equations and Semigroups of Operators | 360 |

15 Viability Kernels and Capture Basins | 385 |

16 FirstOrder Partial Differential Equations | 411 |

Selection of Results | 448 |

Exercises | 470 |

8 Some Approximation Procedures in Spaces of Functions | 167 |

9 Sobolev Spaces of Functions of Several Variables and the Fourier Transform | 187 |

10 Introduction to SetValued Analysis and Convex Analysis | 211 |

488 | |

493 | |

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### Common terms and phrases

associate backward invariant belongs bilinear mapping boundary value problems capture basin Cauchy sequence Chapter characterize closed subset compact operator compact support Consequently contingent cone continuous linear form continuous linear operator convex functions deduce deﬁned deﬁned by Eq Deﬁnition denote dense in H derivative diﬂerential Dom(u Dom(v domain dual duality operator epigraph ﬁnd ﬁrst Fourier transform graph Hence Hilbert space Hilbert-Schmidt operator implies inequality isometry isomorphism Lemma Let f Lipschitz lower semicontinuous map f minimal Moreover nonempty norm obtain operator from H orthogonal projector orthonormal base pivot space pointwise polynomials pre-Hilbert space Proof Let properties Proposition quotients Remark satisﬁes satisfying scalar product Section semigroup sequence of elements set-valued map Sobolev spaces space H Suppose surjective TK(x transpose unbounded operator unique extension vector space vector subspace viability kernel viable under f x e H