## Linear Operators: General theory |

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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

If , however , each of the spaces X1 , . . . , Xn are

be understood , sometimes without explicit mention , that X is the uniquely

determined

= ( ...

If , however , each of the spaces X1 , . . . , Xn are

**Hilbert**spaces then it will alwaysbe understood , sometimes without explicit mention , that X is the uniquely

determined

**Hilbert**space with scalar product ( iv ) ( [ Xp , . - . , X ' n ] , [ ypy • Yn ] )= ( ...

Page 814

Ergebnisse der Math . , V 5 , J . Springer , Berlin , 1942 . Reprinted Edwards Bros

. , Ann Arbor , Mich . , 1947 . 4 . Perturbations des transformations autoadjointes

dans l ' espace de

...

Ergebnisse der Math . , V 5 , J . Springer , Berlin , 1942 . Reprinted Edwards Bros

. , Ann Arbor , Mich . , 1947 . 4 . Perturbations des transformations autoadjointes

dans l ' espace de

**Hilbert**. Comment . Math . Helv . 19 , 347 – 366 ( 1946 - 1947 )...

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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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### 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