## 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 { æn } in an F - space X is called a

corresponds a unique sequence { Q ; } of scalars such that lim lac - - i = 0 . n > 00

8 Let { xn } be a

A sequence { æn } 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 lac - - i = 0 . n > 00

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 VR , Uq . . . , Ug ' s Vg , it is seen that vg is equivalent

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

expansion of the form un = Eb ( ug , v3 ) vg , so that u , is in the closed linear

manifold ...

Thus by forming the chain VR , Uq . . . , Ug ' s Vg , it is seen that vg is equivalent

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

**basis**, the vector u , has anexpansion of the form un = Eb ( ug , v3 ) vg , so that u , is in the closed linear

manifold ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero