## Linear Operators: General theory |

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Page 161

ΕεΣ ů e ba . For a > 0 choose n , so that Ium - Mim Sɛ for m , n 2 ne . Then u ( E ) -

un ( E ) = limm + ( m ( E ) — Ym ( E ) ) , from which it follows that \ u - un Sɛ for n 2

nc . Hence Mon → M , which proves that

ΕεΣ ů e ba . For a > 0 choose n , so that Ium - Mim Sɛ for m , n 2 ne . Then u ( E ) -

un ( E ) = limm + ( m ( E ) — Ym ( E ) ) , from which it follows that \ u - un Sɛ for n 2

nc . Hence Mon → M , which proves that

**ba**(**S**, E , X ) is complete . It follows ...Page 311

Q . E . D . Next we turn to an investigation of the space

The space

is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of ...

Q . E . D . Next we turn to an investigation of the space

**ba**(**S**, E ) . 9 THEOREM .The space

**ba**(**S**, 2 ) is weakly complete . If S is a topological space , the rba ( S )is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of ...

Page 340

16 Let S be a completely regular topological space . Show that C ( S ) is

separable if and only if S is compact and metric . 17 Show that a sequence { 2n }

of elements of

only if ...

16 Let S be a completely regular topological space . Show that C ( S ) is

separable if and only if S is compact and metric . 17 Show that a sequence { 2n }

of elements of

**ba**(**S**, E ) converge weakly to an element à e**ba**(**S**, E ) if andonly if ...

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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