## Linear Operators, Part 1 |

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Page 188

Q.E.D. As in the case of finite measure spaces we shall call the measure space (

S, 2, u) constructed in

product measure space and write (S, 2, u) = P-1 (S1, 2, u). The best known ...

Q.E.D. As in the case of finite measure spaces we shall call the measure space (

S, 2, u) constructed in

**Corollary**6 from the g-finite measure spaces (S, 2, u,) theproduct measure space and write (S, 2, u) = P-1 (S1, 2, u). The best known ...

Page 246

8

notation of

for w” in *** we have wor” – wor, where r =X; i wo (bo)b,. Q.E.D. From

8

**CoRollary**. A finite dimensional normed linear space is reflerive. - PRoof. In thenotation of

**Corollary**7, n n wor = X boCo)"(b), w” = X bow”(b.), 7–1 f=1 - and thusfor w” in *** we have wor” – wor, where r =X; i wo (bo)b,. Q.E.D. From

**Corollary**...Page 422

... William G. Bade, Robert G. Bartle. 11

the linear space &, and let T be a total subspace of £". Then the following

statements are equivalent: (i) f is in T; (ii) f is T-continuous; (iii) S), - {rf(a) = 0} is T-

closed.

... William G. Bade, Robert G. Bartle. 11

**CoRollary**. Let f be a linear functional onthe linear space &, and let T be a total subspace of £". Then the following

statements are equivalent: (i) f is in T; (ii) f is T-continuous; (iii) S), - {rf(a) = 0} is T-

closed.

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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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