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The main result is that if A' is a real Banach space, such that the density character of X* is greater than that of X, then there does not exist any reai-valued Freenet differentiable function on X with bounded nonempty support.

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This is an expository paper in which we study some of the structural and geometric properties of the Banach space l"/c 0 using its identification with C(j8N\N). In particular, it is noted that although LJCQ is not a dual space, its unit ball has an abundance of extreme points. Also, its smooth points are classified and its complemented subspaces are studied. Introduction.The Banach space ljc 0 certainly falls into the category of a "classical Banach space". Not only has it been around since the time of Banach's original monograph (1932) [2], but it is also classical in the sense of Lacey [17] or Lindenstrauss and Tzafriri [18] since it is congruent (isometrically isomorphic) to the space C(/3N\N). However, many of its interesting properties have not been as widely circulated as those of some of the other classical Banach spaces. In this paper, which is of an expository nature, it is our intention to begin to rectify this situation.We will begin with some definitions. /^ is the linear space of bounded sequences of real numbers and c 0 is the subspace of sequences which converge to zero. Both of these spaces, when provided with the supremum norm, ||JC|| = sup|jtj where x = {x n } n ^l9 are Banach spaces. The quotient space IJCQ is the usual linear space of cosets x = x + c 0 , x e / oe . When provided with the quotient norm ||x|| = inf{||jc -y\\ : y e c 0 }, x = x + c 0 , it is a Banach space.Although much is known about quotient spaces in general, this is not the appropriate method for studying ljc$. Indeed, much more is known about C(JT), the space of continuous real valued functions on the compact, Hausdorff space T, and we shall see shortly that IJCQ is congruent to C(/3N\N).Recall that if T is a TychonofF space, (i.e., completely regular and Hausdorff) then its Stone-Cech compactification ßT can be described as AMS (1980) subject classification : 46B25,46B20.

A coincidence theorem of GOEBEL [3] (cf. Theorem 1 below) proved in 1968 has recently received some attention. In particular, OKADA [9], SINGH-VIILENDRA [ 131, and SINOH-KULSHRESTHA [ 121 extended Theorem 1 to L-spaces, 2-metric spaces and metric spaces, respectively. Further, PARK [lo] gave a generalization of it. The purpose of this note is to generalize GOEBEL'S result to a hybrid of multivalued and singlevalued maps satisfying a contractive condition. Consequently, fixed point theorems for multivalued maps of COVITZ-NADLER [ 11, NADLER [7] and SMITHSON [14] are extended. Also coincidence theorems for a pair of multivalued maps are presented. Consistent with [8, p. 6201, X will denote an arbitrary nonempty set, ( M , d ) a metric space, id the identity on M and C L ( M ) (resp. C B ( M ) ) the nonempty closed (resp. closed and bounded) subsets of M . For A , BCCL(M) and E =-0, N(E, A ) = {ZC M : d(x, a) -=E for some aEA) , E A , B = { E > O : A c N ( E , B ) , B c N ( E , A ) } , and, Theorem 1. (GOEBEL [3]). Let S, T : X -, M such that S ( X ) c T ( X ) and T ( X ) is a complete subspace of M. Further, assume there exists k , O e k e 1 such that (1.1) d(Sz, By) s k d ( T x , T y )for all x, yCX. Then S and T have a coincidence, that is, there is zEX such that SZ = Tz.Note that if X = M and T =id above, we have the BANACH contraction principle.The following result generalizes Theorem 1 and the method of proof of this and other results is a variant of the technique employed in [3].