**
R. Khalil, D. Hussein, W. Amin**

##
Geometry of Modulus Spaces

**abstract:**

Let f be a modulus function, i.e., continuous
strictly increasing function on [0, infinity), such that f
(0) = 0, f (1) = 1, and f
(x+y) \leq f (x) + f (y)
for all x, y in [0, infinity). It is the object of this paper to characterize,
for any Banach space X, extreme points, exposed points, and smooth points of the
unit ball of the metric linear space l^{f(X)},
the space of all sequences (x_{n}), x_{n} in X, n = 1, 2, ... ,
for which the sum f (||x_{n}||) is not
infinite. Further, extreme, exposed, and smooth points of the unit ball of the
space of bounded linear operators on l ^{p}, 0 < p < 1, are
characterized.