## Lectures and Exercises on Functional AnalysisThe book is based on courses taught by the author at Moscow State University. Compared to many other books on the subject, it is unique in that the exposition is based on extensive use of the language and elementary constructions of category theory. Among topics featured in the book are the theory of Banach and Hilbert tensor products, the theory of distributions and weak topologies, and Borel operator calculus. The book contains many examples illustrating the general theory presented, as well as multiple exercises that help the reader to learn the subject. It can be used as a textbook on selected topics of functional analysis and operator theory. Prerequisites include linear algebra, elements of real analysis, and elements of the theory of metric spaces. |

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### Contents

1 | |

Normed Spaces and Bounded Operators | 55 |

Banach Spaces and Their Advantages | 125 |

From Compact Spaces to Fredholm Operators | 191 |

Polynormed Spaces Weak Topologies and Generalized | 245 |

functions | 288 |

At the Gates of Spectral Theory | 297 |

Hilbert Adjoint Operators and the Spectral Theorem | 327 |

Fourier Transform | 413 |

455 | |

461 | |

### Common terms and phrases

adjoint operator arbitrary Banach algebra Banach space bijection bilinear Borel bounded functional bounded operator calculus called classical coincides commutative compact operator complete consider contraction operator convergence coproduct Corollary corresponding countable defined Definition denoted dense diagram element epimorphism Example Exercise exists fact family of prenorms finite finite-dimensional formula Fourier transform Fredholm operator functional analysis functor Hausdorff Hence Hilbert spaces Hint homomorphism identity implies integral interval inverse isometric isomorphism isometric operator lemma linear operator linear space locally compact mapping mathematical matrix metric space morphisms non-zero normed space notion numbers objects obviously operator acting operator of multiplication orthogonal orthonormal polynormed space prenormed space Proof properties Proposition quantum space reader respect rest is clear Section selfadjoint selfadjoint operator sequence spectral theorem spectrum subset subspace Suppose surjective tensor product theory topological isomorphism topological space totally bounded unique unitarily unitary equivalence vector zero