Geometry of Modulus Spaces

Let φ be a modulus function, i.e., continuous strictly increasing function on [0,∞), such that φ(0) = 0, φ(1) = 1, and φ(x + y) ≤ φ(x) + φ(y) for all x, y in [0,∞). 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 `(X), the space of all sequences (xn), xn ∈ X, n = 1, 2, . . . , for which ∑ φ(‖xn‖) < ∞. Further, extreme, exposed, and smooth points of the unit ball of the space of bounded linear operators on `, 0 < p < 1, are characterized. 2000 Mathematics Subject Classification: Primary: 47B38; Secondary: 48A65.