## Linear Operators, Part 1 |

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Page 36

The cardinality of a Hamel

is called the dimension of the linear space . This independence is readily proved

if there is a finite Hamel

The cardinality of a Hamel

**basis**is a number independent of the Hamel**basis**; itis called the dimension of the linear space . This independence is readily proved

if there is a finite Hamel

**basis**, in which case the space is said to be finite ...Page 71

A sequence { n } in an F - space X is called a

corresponds a unique sequence { Q ; } of scalars such that lim la – 1 = 0 . 8 Let {

Xn } be a

y ...

A sequence { n } in an F - space X is called a

**basis**for X if to each x € X therecorresponds a unique sequence { Q ; } of scalars such that lim la – 1 = 0 . 8 Let {

Xn } be a

**basis**in the F - space X , and let y be the vector space of all sequencesy ...

Page 254

Thus by forming the chain V 2 , Uq , . . . , Uq ' , Vg , it is seen that ug is equivalent

to v g , and thus that vp is in V . Since { vs } is a

expansion of the form un = { ( ua , vp ) , so that wq is in the closed linear manifold

...

Thus by forming the chain V 2 , Uq , . . . , Uq ' , Vg , it is seen that ug is equivalent

to v g , and thus that vp is in V . Since { vs } is a

**basis**, the vector Wq has anexpansion of the form un = { ( ua , vp ) , so that wq is in the closed linear manifold

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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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 obtained 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