## Linear Operators: General theory |

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Page 188

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

S, E, u) constructed in

product measure space and write (S, E, ft) = The best known example of Theorem

...

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

S, E, u) constructed in

**Corollary**6 from the a-finite measure spaces (S^E^fa) theproduct measure space and write (S, E, ft) = The best known example of Theorem

...

Page 246

8

notation of

have x**x* —x*x, where a; = 2<=ia;**(^*)''r- Q.E.D. From

...

8

**Corollary**. A finite dimensional normed linear space is reflexive. Proof. In thenotation of

**Corollary**7, x*x = J b*(x)x*(bi), x* = f <=i <=i and thus for x** in X** wehave x**x* —x*x, where a; = 2<=ia;**(^*)''r- Q.E.D. From

**Corollary**7 it follows that...

Page 422

11

total subspace of £+. Then the following statements are equivalent: (i) f is in r-, (ii)

/ is r-continuous; (iii) S^f = {x\f{x) = 0} is r-closed. Proof. By Theorem 9, (i) is ...

11

**Corollary**. Let f be a linear functional on the linear space 3E, and let r be atotal subspace of £+. Then the following statements are equivalent: (i) f is in r-, (ii)

/ is r-continuous; (iii) S^f = {x\f{x) = 0} is r-closed. Proof. By Theorem 9, (i) is ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero