## Linear Operators: General theory |

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Page 424

Since each projection is a continuous map . each o the sets A ( 1 , y ) and B ( 1 , I

) is

THEOREM . ( Alaoglu ) The

Since each projection is a continuous map . each o the sets A ( 1 , y ) and B ( 1 , I

) is

**closed**. Hence tK = n sexdir , y cn2e9 , 76x B ( . , I ) is also**closed**. Q . E . D . 2THEOREM . ( Alaoglu ) The

**closed**unit sphere in the conjugate space X * of ...Page 429

( Krein - Šmulian ) A convex set in X * is X -

with every positive multiple of the

This follows from the preceding theorem and Corollary 2.14 . Q.E.D.

COROLLARY .

( Krein - Šmulian ) A convex set in X * is X -

**closed**if and only if its intersectionwith every positive multiple of the

**closed**unit sphere of X * is X -**closed**. Proof .This follows from the preceding theorem and Corollary 2.14 . Q.E.D.

COROLLARY .

Page 488

If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a

range , then UX = Y . Proof . Let 0 + y e Y and define I = { y * y * € Y * , y * y = 0 } .

Then I ' is Y -

and ...

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 define I = { y * y * € Y * , y * y = 0 } .

Then I ' is Y -

**closed**in Y * . Suppose , for the moment , that U * l ' is X -**closed**and ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero