## Linear Operators: General theory |

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Page 34

The binary operation u is often written as u ( a , b ) = ab , and , when this notation

is used , it is

. The product ab is required to satisfy the following conditions : ( i ) a ( bc ) ...

The binary operation u is often written as u ( a , b ) = ab , and , when this notation

is used , it is

**called**multiplication . The element ab is**called**the product of a and b. The product ab is required to satisfy the following conditions : ( i ) a ( bc ) ...

Page 35

A mapping h : A + B between groups A and B is

= h ( a ) h ( b ) . A one - to - one homomorphism is

→ B is an isomorphism and if h ( A ) = B , then A and B are said to be ...

A mapping h : A + B between groups A and B is

**called**a homomorphism if h ( ab )= h ( a ) h ( b ) . A one - to - one homomorphism is

**called**an isomorphism . If h : A→ B is an isomorphism and if h ( A ) = B , then A and B are said to be ...

Page 38

Since there is a one - to - one linear map between the spaces M ; and Xi , the

space X is often

subspace of the vector space X over the field Ø , the factor space X / M is the set

of cosets of ...

Since there is a one - to - one linear map between the spaces M ; and Xi , the

space X is often

**called**the direct sum of the spaces X1 , . . . , Xn . If M is asubspace of the vector space X over the field Ø , the factor space X / M is the set

of cosets of ...

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero