## Linear Operators, Part 1 |

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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 ) ...

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 ) ...

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 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

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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### 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