## 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 ( 0 , x

) is

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

of ...

Since each projection is a continuous map , each o the sets A ( x , y ) and B ( 0 , x

) is

**closed**. Hence tK = n , vex A ( x , y ) n naeg , xe x B ( a , x ) is also**closed**.Q.E.D. THEOREM . ( Alaoglu ) The

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

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

define I ...

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 #ye Y anddefine I ...

Page 489

since the range of U * is

restriction of y * to Z , then x * U * z * . Hence , the range of U * is also

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

Q.E.D. 5 ...

since the range of U * is

**closed**, a * U * y * for some y * € Y * . If z * is therestriction of y * to Z , then x * U * z * . Hence , the range of U * is also

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

**closed**range .Q.E.D. 5 ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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