## 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 , x

) is

D . 2 THEOREM . ( Alaoglu ) The

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

) is

**closed**. Hence tK = ngxex A ( x , y ) o nae , rex B ( a , x ) is also**closed**. Q . E .D . 2 THEOREM . ( Alaoglu ) The

**closed**unit sphere in the conjugate space X ...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 # ye Y and define I = { y * y * e 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 # ye Y and define I = { y * y * e Y * , y * y = 0 }

. Then I ' is Y -

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

Page 489

since the range of U * is

restriction of y * to 3 , then w * = U * * * . Hence , the range of U * is also

follows from the previous lemma that U _ X = UX = 3 . Hence , U has a

range .

since the range of U * is

**closed**, * = U * y * for some y * e Y * . If z * is therestriction of y * to 3 , then w * = U * * * . Hence , the range of U * is also

**closed**. Itfollows from the previous lemma that U _ X = UX = 3 . Hence , U has a

**closed**range .

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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