## Linear Operators: General theory |

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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 { xn } in an F - space X is called a

corresponds a unique sequence { a ; } of scalars such that lim la Σ αγα , = 0 . Σ α i

= 1 8 Let { xn } be a

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

**basis**for X if to each x € * therecorresponds a unique sequence { a ; } of scalars such that lim la Σ αγα , = 0 . Σ α i

= 1 8 Let { xn } be a

**basis**in the F - space X , and let y be the vector space of all ...Page 254

Thus by forming the chain UB , UG , ... , Uz ' , Up . it is seen that vg is equivalent to

vg , and thus that up is in V. Since { vp } is a

expansion of the form ug = { ( uq , UB ) UBY Ex ( uq , vBlog , so that u , is in the

closed ...

Thus by forming the chain UB , UG , ... , Uz ' , Up . it is seen that vg is equivalent to

vg , and thus that up is in V. Since { vp } is a

**basis**, the vector Wq has anexpansion of the form ug = { ( uq , UB ) UBY Ex ( uq , vBlog , so that u , is in the

closed ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen 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