## Linear Operators: General theory |

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Page 244

Finite Dimensional Spaces The space En , as will be seen presently , is the

prototype of all n - dimensional normed linear spaces , and

observed first that En is a B - space . The Minkowski inequality ( III.3.3 ) shows En

to be ...

Finite Dimensional Spaces The space En , as will be seen presently , is the

prototype of all n - dimensional normed linear spaces , and

**hence**it should beobserved first that En is a B - space . The Minkowski inequality ( III.3.3 ) shows En

to be ...

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 * ( Tr ) / < € , so that Tx e N ( 0 ; yi , ... , 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

...

Page 441

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

subset of co ( Q ) .

easy induction on Lemma 2 . 5 . It follows readily that p has the form p = { - 1 a ; k

...

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

**hence**a compact ,subset of co ( Q ) .

**Hence**CO ( Q ) = Co ( KU . . . UKn ) = co ( KU . . . UKn ) , by aneasy induction on Lemma 2 . 5 . It follows readily that p has the form p = { - 1 a ; k

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero