## Linear Operators: General theory |

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Page 65

14

x*\ = 1 and x*x = \x\. Proof. Apply Lemma 12 with 3) = 0. The x* required in the

present

14

**Corollary**. For every x ^ 0 in a normed linear space X, there is an x* e X* with \x*\ = 1 and x*x = \x\. Proof. Apply Lemma 12 with 3) = 0. The x* required in the

present

**corollary**may then be defined as \x\ times the x* whose existence is ...Page 188

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

S. E. p) constructed in

product measure space and write (S,E,p) = P7-i (Si, zt,tii)- The best known ...

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

S. E. p) constructed in

**Corollary**6 from the cr-finite measure spaces (SjyEjtfif) theproduct measure space and write (S,E,p) = P7-i (Si, zt,tii)- The best known ...

Page 422

11

subspace of X+. Then the following statements are equivalent: (i) f is in /; (ii) / is r-

continuous; (iii) ?Qt = {x\f(x) = °} r-closed. Proof. By Theorem 9, (i) is equivalent ...

11

**Corollary**. Let f be a linear functional on the linear space X, and let r be a totalsubspace of X+. Then the following statements are equivalent: (i) f is in /; (ii) / is r-

continuous; (iii) ?Qt = {x\f(x) = °} r-closed. Proof. By Theorem 9, (i) is equivalent ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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