## Linear Operators, Part 1 |

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Page 421

Unl = X. 0.; 'si. i=1

functional in T is T-continuous, by Lemma 8. / Conversely, let g # so be a linear

functional on 3 which is I'-continuous. There exists a |T-neighborhood N(0; fl. ...,

fa; ...

Unl = X. 0.; 'si. i=1

**Hence**g(a) a e 3. Q.E.D. PRoof of THEOREM 94. Everyfunctional in T is T-continuous, by Lemma 8. / Conversely, let g # so be a linear

functional on 3 which is I'-continuous. There exists a |T-neighborhood N(0; fl. ...,

fa; ...

Page 441

Then K, is a closed, and

U... U.K.) = co(K, U. . . U.K.). by an easy induction on Lemma 2.5. It follows readily

that p has the form p = X_1 a.k, a, 2 0, X_1a, + 1, k, e K, and, since p is extremal, ...

Then K, is a closed, and

**hence**a compact, subset of co(Q).**Hence**co(Q) = co(K,U... U.K.) = co(K, U. . . U.K.). by an easy induction on Lemma 2.5. It follows readily

that p has the form p = X_1 a.k, a, 2 0, X_1a, + 1, k, e K, and, since p is extremal, ...

Page 485

Lemma 7 that T+* is continuous relative to the 3", )*** topologies in 3:44, J)**,

respectively. If S, S** are the closed unit spheres in 3, 3.4°, respectively, and if x

is ...

**Hence**T4 is weakly compact. Conversely, if To is weakly compact, it follows fromLemma 7 that T+* is continuous relative to the 3", )*** topologies in 3:44, J)**,

respectively. If S, S** are the closed unit spheres in 3, 3.4°, respectively, and if x

is ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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