## Linear Operators: General theory |

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Page 421

By II.3.11 , it can be extended to a linear functional y , on En . By IV.3.7 , y , has the form The map y n [ yı , ... , yn ] Σα ; Jii = 1

By II.3.11 , it can be extended to a linear functional y , on En . By IV.3.7 , y , has the form The map y n [ yı , ... , yn ] Σα ; Jii = 1

**Hence**n g ( x ) = ant . ( x ) , X e X. Q.E.D. + PROOF OF THEOREM 9. Every functional in I ' is I ...Page 441

Put co ( ( qi + UnQ ) C9i + U . Then K ; is a closed , and

Put co ( ( qi + UnQ ) C9i + U . Then K ; is a closed , and

**hence**a compact , subset of co ( Q ) .**Hence**CO ( Q ) = co ( K , U ... UK ) = co ( KU ... UK ) . by an easy induction on Lemma 2.5 . It follows readily that р has the { = 1 a ...Page 485

**Hence**T * is weakly compact . Conversely , if T * is weakly compact , it follows 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 ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian 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