## 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 ( be ) = ( ab )

...

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 ( be ) = ( ab )

...

Page 35

A one - to - one homomorphism is

isomorphism and if h ( A ) = B , then A and B are said to be isomorphic , or A is

said to be isomorphic with B. An isomorphism of a group G with itself is

...

A one - to - one homomorphism is

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

said to be isomorphic with B. An isomorphism of a group G with itself is

**called**an...

Page 38

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

space X is often

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

M ...

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

space X is often

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

M ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element 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 meaning measure space metric neighborhood norm operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero