## 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

A → 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 Mi and Xin 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 Mi and Xin the

space X is often

**called**the direct sum of the spaces X7 , . . . , 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

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

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