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That for each x~D G , PG(X) I, is or not superfluous. 2) there exists in G. Then by Theorem PG(X))=T(x,g). (3) we have for each g~G that ~ ( g ) ~ ( x , g ) = 0 ~eAsp{G,x } (x) (~) in the cone PG(X) sp{G,x}" we have for each geG that 0=dist(-g,cone Since by G and x. subspace with property s~ace E, and let x~p~I(0)f%SE . _ fo#AE(X) such that the restriction foiG=0. Suppose that cone PG(X) is norm-dense ~(g)=0. (~) in E, which the proof. is proximinal By lg+(l-l)goePG(X), ~(e)=l Igo+~(g-go) II is convex (8) we get: contradicting completes (6).

Math. J~rusalem, 13,(1964). ON S U M M A B I L I T Y IN CONJUGATE BANACH SPACES ~) R. Brigola NWF I - Mathematik, Universit~t R e g e n s b u r g 8400 Regensburg, Germany 0. 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.

40 THEOREM 3. Proo~ Let If C By a s t a n d a r d x be a n o n - e m p t y application s e n s e of i n c l u s i o n ) T(CI) c CI. is a s u p r e m u m of the C1. minimality point of Further, of points For each convex and CI has (FPP*). t, that itself argument C1 c C1 with A non-expansive. , ilXn-XmEl > e under T convex a n d so b y then every and Milman we may extract where we have it f o l l o w s w*-compact than one point, of B r o d s k i i and by a multiplication we h a v e itself is a n o n - e m p t y is i n v a r i a n t contains a n d b y an a r g u m e n t r(C I) = m i n { r ( z ) : x 6 C1 C1 by a standard (Xn) (WUKK*) that of Now suppose of erality w*-compact A is d i a m e t r a l a sequence subset l e m m a we m a y r e p l a c e functions center definition convex x := Sup{l]z-yli: y E C I} A = C 1.