## 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 ( « , x

) is

Q.E.D. 2 THEOREM . ( Alaoglu ) The

...

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

) is

**closed**. Hence tK = n x , ve ¥ A ( x , y ) nnaeyrex B ( a , x ) is also**closed**.Q.E.D. 2 THEOREM . ( Alaoglu ) The

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

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

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

and ...

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. Ux = PROOF . Let 0 #ye Yand ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

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