## Linear Operators: General theory |

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Page 329

The Space

subsets, and a scalar valued countably additive set function fx on E. The symbol

The Space

**TM**(**S**, E, /x) We shall be concerned here with a set S, a a-field E of itssubsets, and a scalar valued countably additive set function fx on E. The symbol

**TM**(**S**, E, fx) 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 An of [0, 1] of length less than 1/n and such that x* does not

...

linear functional on

**TM**(**S**, 27, //) which is not identically zero, then, since linearcombinations of characteristic functions of intervals are dense in

**TM**(**S**, 27, fi),there is a subinterval An of [0, 1] of length less than 1/n and such that x* does not

...

Page 333

6e denumerable sets and for each a e Ax let Ta be a continuous linear map from

an F-space X into the space

measurable functions. Suppose, that (i) for each x in X sup \Ta(x, s)\ < oo, atAl

almost ...

6e denumerable sets and for each a e Ax let Ta be a continuous linear map from

an F-space X into the space

**TM**(**S**, E, fi) of real or complex valued totallymeasurable functions. Suppose, that (i) for each x in X sup \Ta(x, s)\ < oo, atAl

almost ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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### Common terms and phrases

Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero