## Linear Operators: General theory |

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Page 421

7 , 4 , has the form Vi lyp - - „ Yn ] = { x : Si

. E . D . PROOF OF THEOREM 9 . Every functional in I ' is I - continuous , by

Lemma 8 . Conversely , let g + 0 be a linear functional on X which is I ' -

continuous .

7 , 4 , has the form Vi lyp - - „ Yn ] = { x : Si

**Hence**8 ( x ) = { viti ( x ) , X e X . i = 1 Q. E . D . PROOF OF THEOREM 9 . Every functional in I ' is I - continuous , by

Lemma 8 . Conversely , let g + 0 be a linear functional on X which is I ' -

continuous .

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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 e N ( 0 ; yt , . . . , Y , € ) . Therefore , Tis 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 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect 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