## Theory of Linear OperationsThis classic work by the late Stefan Banach has been translated into English so as to reach a yet wider audience. It contains the basics of the algebra of operators, concentrating on the study of linear operators, which corresponds to that of the linear forms a1x1 + a2x2 + ... + anxn of algebra. The book gathers results concerning linear operators defined in general spaces of a certain kind, principally in Banach spaces, examples of which are: the space of continuous functions, that of the pth-power-summable functions, Hilbert space, etc. The general theorems are interpreted in various mathematical areas, such as group theory, differential equations, integral equations, equations with infinitely many unknowns, functions of a real variable, summation methods and orthogonal series. A new fifty-page section (``Some Aspects of the Present Theory of Banach Spaces'') complements this important monograph. |

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### Contents

1 | |

5 | |

13 | |

17 | |

23 | |

Chapter IV Normed spaces | 33 |

Chapter V Banach spaces | 49 |

Chpter VI Compact operators | 59 |

Appendix Weak convergence in Banach spaces | 127 |

Remarks | 137 |

157 | |

Some aspects of the present theory of Banach spaces | 161 |

Introduotion | 163 |

Chapter I | 165 |

Chapter II Local properties of Banach spaces | 169 |

Chapter III The approximation property and bases | 179 |

Chapter VII Biorthogonal sequences | 65 |

Chapter VIII Linear functionals | 71 |

Chapter IX Weakly convergent sequences | 81 |

Chapter X Linear functional equations | 89 |

Chapter XI Isometry equivalence isomorphism | 101 |

Chapter XII Linear dimension | 117 |

Chapter IV | 189 |

Chapter V Classical Banach spaces | 195 |

Chapter VI | 209 |

217 | |

Additional Bibliography | 233 |

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### Common terms and phrases

adjoint Amer arbitrary B-measurable Banach space Bessaga bijectively bounded linear functional bounded linear operator Chapter closed linear subspace codomain compact linear operator complemented subspace complete metric space consequently continuous function convergent sequence converges weakly denote dual space elements equations equivalent example exists a sequence f defined F-space Figiel finite following theorem follows from theorem functional f functions defined given Grothendieck Hausdorff space Hilbert space homeomorphic hypothesis implies inequality infinite-dimensional Banach space isometrically isomorphic Kadec lemma linear space linear subspace mapping Mazur measure n+oo natural number necessary and sufficient norm norm-bounded Orlicz pair Pelczyński proved real numbers reflexive regularly closed Remark Riesz Rosenthal satisfying the conditions separable Banach space sequence of functions sequence of numbers sequence xn set G space E space LP Studia Math subset summable theory topological transfinite unconditional basis vector space weak convergence whence