## Linear Operators: General theory |

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Page 421

7 , y , has the form Valyv , . . , Yn ] = { Xi Yi

. E . D . PROOF OF THEOREM 9 . Every functional in l ' is - continuous , by

Lemma 8 . Conversely , let g + 0 be a linear functional on X which is I ' -

continuous .

7 , y , has the form Valyv , . . , Yn ] = { Xi Yi

**Hence**g ( x ) = { vif : ( a ) , X e X . i = 1 Q. E . D . PROOF OF THEOREM 9 . Every functional in l ' is - continuous , by

Lemma 8 . Conversely , let g + 0 be a linear functional on X which is I ' -

continuous .

Page 441

Put K ; = collqi + U ) n Q ) ÇqitU . Then K ; is a closed , and

subset of co ( Q ) .

an easy induction on Lemma 2 . 5 . It follows readily that p has the form p = { 1 –

1a ...

Put K ; = collqi + U ) n Q ) ÇqitU . Then K ; is a closed , and

**hence**a compact ,subset of co ( Q ) .

**Hence**CO ( Q ) = co ( K , U . . . UKN ) = co ( K , U . . . UKn ) , byan easy induction on Lemma 2 . 5 . It follows readily that p has the form p = { 1 –

1a ...

Page 485

from Lemma 7 that T * * is continuous relative to the X * , Y * * * topologies in X * *

, Y * * , respectively . If S , S * * are the closed unit spheres in X , X * * ,

respectively ...

**Hence**7 * is weakly compact . Conversely , if T * is weakly compact , it followsfrom Lemma 7 that T * * is continuous relative to the X * , Y * * * topologies in X * *

, Y * * , respectively . If S , S * * are the closed unit spheres in X , X * * ,

respectively ...

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

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