## Linear Operators: General theory |

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Page 421

<=i

"-continuous, by Lemma 8. Conversely, let g ^0 be a linear functional on i' which

is /'-continuous. There exists a /-neighborhood N(0; fv . . ., /„; e) which is mapped ...

<=i

**Hence**n sW = I«i/;(4 ««3£. Q.E.D. Proof of Theorem 9. Every functional in r is /"-continuous, by Lemma 8. Conversely, let g ^0 be a linear functional on i' which

is /'-continuous. There exists a /-neighborhood N(0; fv . . ., /„; e) which is mapped ...

Page 423

Then \x*\ g \y*\ □ \T\, and so x* e X*. If x e N(0; x* x*, e)'then \x*(x)\ < e.

Tx)\ < e, so that Tx e 2V(0; y*, . . ., y*, e). Therefore, Tis weakly continuous at the

origin, and

continuous ...

Then \x*\ g \y*\ □ \T\, and so x* e X*. If x e N(0; x* x*, e)'then \x*(x)\ < e.

**Hence**\yf(Tx)\ < e, so that Tx e 2V(0; y*, . . ., y*, e). Therefore, Tis weakly continuous at the

origin, and

**hence**at every point. Conversely, suppose that T is weaklycontinuous ...

Page 441

covering of Q; let {qi+U}, i = 1, . . ., n, be a finite subcovering. Put Kf = co((9,-f£/)n

Q) Q qi+U. Then K{ is a closed, and

Q) = co^ u • • • U K») = co^ U • • • U Kn), by an easy induction on Lemma 2.5.

covering of Q; let {qi+U}, i = 1, . . ., n, be a finite subcovering. Put Kf = co((9,-f£/)n

Q) Q qi+U. Then K{ is a closed, and

**hence**a compact, subset of co(Q).**Hence**co(Q) = co^ u • • • U K») = co^ U • • • U Kn), by an easy induction on Lemma 2.5.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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