## Linear Operators, Part 1 |

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Page 329

The Space

subsets, and a scalar valued countably additive set function u on 2. The symbol

The Space

**TM**(**S**, X, u) We shall be concerned here with a set S, a g-field 2 of itssubsets, and a scalar valued countably additive set function u on 2. The symbol

**T.M.**(**S**, 2, u) will be used for the set of all scalar valued functions on S which are ...Page 330

linear functional on

combinations of characteristic functions of intervals are dense in

there is a subinterval A, of [0, 1] of length less than 1/m and such that a* does not

...

linear functional on

**TM**(**S**, X, u) which is not identically zero, then, since linearcombinations of characteristic functions of intervals are dense in

**TM**(**S**, X, u),there is a subinterval A, of [0, 1] of length less than 1/m and such that a* does not

...

Page 333

Let A, D A, D ... be denumerable sets and for each a e A, let T. be a continuous

linear map from an F-space & into the space

totally measurable functions. Suppose that (i) for each æ in 3: sup T. (a, s) < 00, ...

Let A, D A, D ... be denumerable sets and for each a e A, let T. be a continuous

linear map from an F-space & into the space

**TM**(**S**, 2, u) of real or compler valuedtotally measurable functions. Suppose that (i) for each æ in 3: sup T. (a, s) < 00, ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

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