## Linear Operators: General theory |

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Page 65

14

= 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, Z, fi) constructed in

product measure space and write (S,Z,/i) = PI* The best known example of ...

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

S, Z, fi) constructed in

**Corollary**6 from the a-finite measure spaces (Sj, Zit fii) theproduct measure space and write (S,Z,/i) = PI* The best known example of ...

Page 246

The following

. 7

the functional^ b*, i = 1 n, defined by the equations n form a Hamel basis for ...

The following

**corollary**was established during the first part of the preceding proof. 7

**Corollary**. If {bv . . ., bn} is a Hamel basis for the normed linear space X thenthe functional^ b*, i = 1 n, defined by the equations n form a Hamel basis for ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero