## Linear Operators: General theory |

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Page 421

Ini ) which is mapped by g into the unit sphere of Ø . For e X * let S = { x \ | ( x ) = 0

} , and suppose that x , en - 10 ,; Then x , € N ( 0 ; f1 , ... , In , € ) , and

) / < 1. Since n - 1.0 , is a linear space , it contains mx , for every integer m .

Ini ) which is mapped by g into the unit sphere of Ø . For e X * let S = { x \ | ( x ) = 0

} , and suppose that x , en - 10 ,; Then x , € N ( 0 ; f1 , ... , In , € ) , and

**hence**g ( x )) / < 1. Since n - 1.0 , is a linear space , it contains mx , for every integer m .

Page 423

continuous at the origin , and

is weakly continuous , and y * € Y * . Then y * T is a linear functional on X which is

...

**Hence**\ y * ( Tx ) ) < € , so that Tx € N ( 0 ; y * , ... Y * , £ ) . Therefore , T'is weaklycontinuous at the origin , and

**hence**at every point . Conversely , suppose that Tis weakly continuous , and y * € Y * . Then y * T is a linear functional on X which is

...

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from Lemma 7 that T ** is continuous relative to the X * , Y *** topologies in X ** ,

Y ** , respectively . If S , S ** are the closed unit spheres in X , X ** , respectively ...

**Hence**T * is weakly compact . Conversely , if T * is weakly compact , it followsfrom Lemma 7 that T ** is continuous relative to the X * , Y *** topologies in X ** ,

Y ** , respectively . If S , S ** are the closed unit spheres in X , X ** , respectively ...

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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