## Linear Operators, Part 1 |

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Page 243

The norm in H is x = ( 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

The norm in H is x = ( 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****space**...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 X , ... , Xn

are

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 X , ... , Xn

are

**Hilbert spaces**then it will always be understood , sometimes without explicit ...Page 851

... in B - spaces , II.3.4 ( 59 ) discussion of , ( 82–83 ) in F - spaces , II.1.14–16 ( 54

) definition , ( 36 ) extensions of , VI.2.5 ... of a linear space , II.3.1 ( 59 )

Orthocomplement of a set in

IV.4.4 ...

... in B - spaces , II.3.4 ( 59 ) discussion of , ( 82–83 ) in F - spaces , II.1.14–16 ( 54

) definition , ( 36 ) extensions of , VI.2.5 ... of a linear space , II.3.1 ( 59 )

Orthocomplement of a set in

**Hilbert space**, definition , IV.4.3 ( 249 ) properties ,IV.4.4 ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero