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 ℓ
ϕ
(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 ℓ𝑝, 0 < 𝑝 < 1, are characterized.
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