## Linear Operators: General theory |

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Page 421

functional in l ' is ' - continuous , by Lemma 8 . Conversely , let g + o be a linear

functional on X which is I ' - continuous . There exists a I - neighborhood N ( 0 ; 1 .

.. , İn ; E ) ...

**Hence**n X e X. g ( x ) = { viti ( x ) , Q.E.D. PROOF OF THEOREM 9. Everyfunctional in l ' is ' - continuous , by Lemma 8 . Conversely , let g + o be a linear

functional on X which is I ' - continuous . There exists a I - neighborhood N ( 0 ; 1 .

.. , İn ; E ) ...

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Put K ; = co ( ( qi + U ) NQ ) Çqi + U . Then K ; is a closed , and

subset of co ( Q ) .

easy induction on Lemma 2.5 . It follows readily that p has the { -1a ; k , a ; 20 ...

Put K ; = co ( ( qi + U ) NQ ) Çqi + U . Then K ; is a closed , and

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

**Hence**CO ( Q ) = co ( K , U ... UKN ) = co ( K , U ... UKn ) , by aneasy induction on Lemma 2.5 . It follows readily that p has the { -1a ; k , a ; 20 ...

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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

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad 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 semi-group 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