Some geometric properties of the duals of Cesàro sequence spaces

Abstract The spaces d ( s ) are defined for 0 ≤ s ≤ ∞ . We consider the fundamental geometric properties of the d ( s ) spaces, isomorphic duals of the Cesàro sequence spaces ces ( r ) with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:mfrac> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> $\frac{1}{s}+\frac{1}{r}=1$ . We prove that for 1 ≤ s &lt; ∞ , the Banach spaces d ( s ) are Radon-Riesz spaces that are not rotund or smooth. Moreover, we show that the Banach lattice d (1) has Schur’s property, just as ℓ 1 does. Finally, a characterization of norm totally bounded subsets of d (1) is also given.