## Linear Operators: General theory |

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Page 243

The norm in H is 1x1 = ( x , x ) 1 / 2 Remark .

set of abstract axioms . It is noteworthy that some of the concrete spaces defined

above satisfy these axioms , and hence are special cases of abstract Hilbert ...

The norm in H is 1x1 = ( x , x ) 1 / 2 Remark .

**Hilbert space**has been defined by aset of abstract axioms . It is noteworthy that some of the concrete spaces defined

above satisfy these axioms , and hence are special cases of abstract Hilbert ...

Page 247

closely related , especially in its elementary geometrical aspects , to the

Euclidean or finite dimensional unitary spaces . It is not immediate from the

definition ( 2 .

**Hilbert Space**Of the infinite dimensional B - spaces ,**Hilbert space**is the mostclosely related , especially in its elementary geometrical aspects , to the

Euclidean or finite dimensional unitary spaces . It is not immediate from the

definition ( 2 .

Page 256

Whenever the direct sum of normed linear spaces is used as a normed space ,

the norm will be explicitly mentioned . If , however , each of the spaces X1 , . . . ,

Xn are

explicit ...

Whenever the direct sum of normed linear spaces is used as a normed space ,

the norm will be explicitly mentioned . If , however , each of the spaces X1 , . . . ,

Xn are

**Hilbert spaces**then it will always be understood , sometimes withoutexplicit ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero