## Linear Operators: General theory |

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Page 424

Since each projection is a continuous map , each o the sets A ( x , y ) and B ( a ,

æ ) is

E . D . 2 THEOREM . ( Alaoglu ) The

...

Since each projection is a continuous map , each o the sets A ( x , y ) and B ( a ,

æ ) is

**closed**. Hence tK = nt , vex A ( x , y ) onde pcex B ( « , x ) is also**closed**. Q .E . D . 2 THEOREM . ( Alaoglu ) The

**closed**unit sphere in the conjugate space X...

Page 429

Q . E . D . 7 THEOREM . ( Krein - Smulian ) A convex set in X * is X -

only if its intersection with every positive multiple of the

is X -

Q . E . D . 7 THEOREM . ( Krein - Smulian ) A convex set in X * is X -

**closed**if andonly if its intersection with every positive multiple of the

**closed**unit sphere of X *is X -

**closed**. Proof . This follows from the preceding theorem and Corollary 2 .Page 488

It follows from the definition of U * that every element in its range satisfies the

stated condition . Q . E . D . 3 LEMMA . If the adjoint of an operator U in B ( X , Y )

is one - to - one and has a

...

It follows from the definition of U * that every element in its range satisfies the

stated condition . Q . E . D . 3 LEMMA . If the adjoint of an operator U in B ( X , Y )

is one - to - one and has a

**closed**range , then UX = Y . PROOF . Let 0 + y e Y and...

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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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### Common terms and phrases

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