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By Author Unknown

Because the first version of this recognized textual content used to be released in 1982, major development has been made within the neighborhood idea of Banach areas. This moment variation has accordingly been pointed out to this point via the addition of a totally new part dedicated to this subject, in addition to a variety of different revisions, an elevated bibliography and a brand new appendix.

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1 - COROLLARY 4. x E Weif , and iA abubbpace 0 6 G 16 o(E/Gl F = G1 - * G ) = o(E, E ) J G l 9 such ends t h e p r o o f o f o u r p r o p o s i t i o n . have : PROOF. 1 , t h e r e i s an x ' E WE ;'! 1,.. ,c 1I . . 'fn x ' E F : t h i s proves t h a t p(W, ;f ) i s contained i n 1'. * . 'fn Therefore, f o r a l l that . f ! ( x ' ) = f . ( x ' ) = fi(X) , E which iA o(E*, E)-cLubed, we * We come back t o t h e s i t u a t i o n o f t h e p r e v i o u s p r o p o s i t i o n , p u t t i n g : we have F 1 = G , G since is , E)-closed.

L e t us now l o o k f o r t h e t r a c e on E o f t h e t o p o l o g y has a meaning, s i n c e ** . A E i s a subspace o f E E*) u(E**, : this fundamental system o f neighbourhoods o f t h e o r i g i n i s , by d e f i n i t i o n , c o n s t i t u t e d b y t h e intersection with E of t h e s e t s ( 2 ) . ,5n * . E E t h i s t o p o l o g y : i t i s t h e weak t o p o l o g y on E From t h e f o r m o f t h e neighbourhoods ( 3 ) , one deduces t h a t i t i s t h e a nequence o(E, E*) ( x ~E )N ~ o f elements o f i f and o n l y i f , f o r every E converges t o a p o i n t C; E E* < xn , 5 > w i t h t h e new n o t a t i o n s , .

Ltn(x)1 be a neighbourhood o f and t h i s proves t h a t t h e t o p o l o g y 1 i s stronger than u ( E l F , F ) a(E, So Q llEIIE* t h e canonical s u r j e c t i v e mapping from {i E = ) . F modulo E l F by llEll(EIF)* d e f i n e a l i n e a r f u n c t i o n a l on i s t h e image by ,tn ;El'. En E F 1" we 1'. i , and i s any element i n i s the class o f b) Let d e f i n e s a l i n e a r f u n c t i o n a l on I ) * i s s t r o n g e r t h a n o(E, E )/F E* spanned by , which F1 and E l , .

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