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

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

**Sample text**

Let x,yeE be such that T(x,y)=l IYl I- Then T(y,x) =] Ixl ]. Indeed, by (3) there exists fo~AE(X) so foEAE(Y). , T(y,x):[ ]xl I. such thatllyl [=T(x,y)=fo(y) (3), we have and [Ixl [=fo(X)~T(y,x)g[[x[ I, THEOREM 2. Let G be a linear subspace of the normed linear space E. Consider the following statements: i) G is semi-Cheb[shev and has property ii) For each X,PGI(0) iii) G=n P (') i_~nE. and each gaG we have T(x,g)=[ Igl !. x)(0) DGIx P I iv) PG 1 (0)=~{PN l(g) (0) ]geG} We have i ) ~ i i ) , ii) ~=~iii), ii) ~ i v ) .

Summary G e n e r a l i z a t i o n s of the B a n a c h - S a k s p r o p e r t y were used by several authors to characterize reflexive Banach spaces (cf. [Ii~, [12~, and [16~). We give a c h a r a c t e r i z a t i o n of separable conjugate Banach spaces by a similar summability condition. As a consequence, we obtain analogous c h a r a c t e r i z a tions of separable second conjugate Banach spaces and of q u a s i - r e f l e x i v e spaces. N o n s e p a r a b l e conjugate Banach spaces p o s s e s s i n g a smooth predual are also c h a r a c t e r i z e d in terms of a s u m m a b i l i t y condition.

C) It is easily seen that a slight change in the norm of a u n i f o r m l y c o n v e x space (depending on the modulus o f convexity) T h e o r e m i, any such space has In m a n y examples ing that the (FPP) w - [w e] said to satisfy the w or (FPP e) Recall that a [dual] space is O p i a l condition if [Xn ~ x 0] xn £ X (n = 1,2 .... ), implies liminftl x0-xnll < liminfll X-Xnll for all and that a [dual] space s a t i s f y i n g the [(FPPe)] (cf. [3]). w [ w e] x ~ x 0, Opial c o n d i t i o n has (FPP) The following example shows that sometimes we can get re- sults even in cases w h e r e the Opial c o n d i t i o n is not satisfied.