## Linear Operators, Part 1 |

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Page 436

Show that if 3 contains a non-void T-open bounded (II.1.7) set, then 3: must be

finite dimensional. 6 Let 3: be a B-space, and 3., a

** topology of 3:1 is the same as the relative 3" topology of 3:1. 7 Let 3: be a

linear ...

Show that if 3 contains a non-void T-open bounded (II.1.7) set, then 3: must be

finite dimensional. 6 Let 3: be a B-space, and 3., a

**subspace**of 3. Show that the** topology of 3:1 is the same as the relative 3" topology of 3:1. 7 Let 3: be a

linear ...

Page 504

Then there exists a closed separable

m, n = 1, 2, ..., belong to 3 and satisfy the relations ran = 1 and a' (ann) = ( - }) |r.

Then there exists a closed separable

**subspace**) of t such that J is equivalent to a**subspace**of Ş)*. PRoof. Let {ai} be a countable dense subset of Jo, and let {ran),m, n = 1, 2, ..., belong to 3 and satisfy the relations ran = 1 and a' (ann) = ( - }) |r.

Page 513

then U has a continuous inverse. 16 If \) is a closed

Qt is a finite dimensional

a closed

.

then U has a continuous inverse. 16 If \) is a closed

**subspace**of a B-space andQt is a finite dimensional

**subspace**, then \) (P 92 is a closed**subspace**. If \) B J isa closed

**subspace**, and 97 is finite dimensional, it does not follow that \) is closed.

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero