## Linear Operators: General theory |

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Page 243

The norm in § is \x\ = (x, x)lli. Remark.

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

satisfy these axioms, and hence are special cases of abstract

The norm in § is \x\ = (x, x)lli. Remark.

**Hilbert space**has been defined by a set ofabstract axioms. It is noteworthy that some of the concrete spaces defined above

satisfy these axioms, and hence are special cases of abstract

**Hilbert space**...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 (2.26) ...

**Hilbert Space**Of the infinite dimensional J5-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 (2.26) ...

Page 256

If, however, each of the spaces £j, . . ., Xn are Hilbcrt spaces then it will always be

understood, sometimes without explicit mention, that 3£ is the uniquely

determined

y{){, ...

If, however, each of the spaces £j, . . ., Xn are Hilbcrt spaces then it will always be

understood, sometimes without explicit mention, that 3£ is the uniquely

determined

**Hilbert space**with scalar product X (iv) ([xv . . ., x„], [yv. ..,»/„]) = 2 (a;,-,y{){, ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 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 |

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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