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By Jonathan Arazy (auth.), Albrecht Pietsch, Nicolae Popa, Ivan Singer (eds.)

ISBN-10: 3540122982

ISBN-13: 9783540122982

ISBN-10: 3540398775

ISBN-13: 9783540398776

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Read or Download Banach Space Theory and its Applications: Proceedings of the First Romanian-GDR Seminar Held at Bucharest, Romania, August 31 – September 6, 1981 PDF

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Extra info for Banach Space Theory and its Applications: Proceedings of the First Romanian-GDR Seminar Held at Bucharest, Romania, August 31 – September 6, 1981

Example text

COROLLARY 2. Let E be a normed let gaS E be such that N(g)=sp{g}. PROOF. Use Theorem linear space with property Then G=sp{g} 2, iv)~->i), observing (A) and is a semi L-summand that N(Ix)=N(x) in E. for each I~0 and that N(0)={0}. 8) proved the following space of all real-valued Hausdorff space Q). Let G be a closed L-summand in E if and onl[ if G={0}, For E=C(Q) sp{x}, where and xeE we have crit x={q~Ql semi L-summand in C(Q), the above mentioned For G ~ E , mations Notice follows subspace Let E=C(Q) on a compact of E.

C o n s e q u e n t l y JX,V' is an isometry from X onto V' As a consequence of theorem 2, we obtain the r e f l e x i v i t y of some special classes of B a n a c h spaces. The following facts are known and u s u a l l y proved by an a p p l i c a t i o n of the deep theorem of R. C. James E9~, (cf. E6~). Here, these results e s s e n t i a l l y appear as an easy c o n s e q u e n c e of the B i s h o p - P h e l p s Corollary 5 theorem. Let X be a B a n a c h space. I) If X' is very smooth, then X is reflexive.

Are true. (x) n C of not. c C X consisting Then, by a satisfying l i m d(Xn+ I, co{x I ..... X n }) = d i a m C n~ A n y subsequence of (Xn) again satisfies (i) , so we may, by w e a k compactness, assume w that x ~ x. en a multiplication, we may n w further s i m p l i f y the s i t u a t i o n and assume t h a t x ~ 0 and d i a m C = I. Since n the w e a k and the n o r m c l o s u r e of co(x ) coincide, (1) implies in particular that n lim llx II = I. N o w let e < i and ~ > 0 b e as in the d e f i n i t i o n of (WUKK).

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Banach Space Theory and its Applications: Proceedings of the First Romanian-GDR Seminar Held at Bucharest, Romania, August 31 – September 6, 1981 by Jonathan Arazy (auth.), Albrecht Pietsch, Nicolae Popa, Ivan Singer (eds.)


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