Abstract
Abstract We introduce the notion of a strongly orthogonal set relative to an element in the sense of Birkhoff-James in a normed linear space to find a necessary and sufficient condition for an element x of the unit sphere S X to be an exposed point of the unit ball B X . We then prove that a normed linear space is strictly convex iff for each element x of the unit sphere, there exists a bounded linear operator A on X which attains its norm only at the points of the form λx with λ ∈ S K . MSC:46B20, 47A30.
Highlights
Suppose (X, · ) is a normed linear space over the field K, real or complex
X is said to be strictly convex iff every element of the unit sphere SX = {x ∈ X : x = } is an extreme point of the unit ball BX = {x ∈ X : x ≤ }
The notion of strict convexity plays an important role in the studies of the geometry of Banach spaces
Summary
Suppose (X, · ) is a normed linear space over the field K, real or complex. X is said to be strictly convex iff every element of the unit sphere SX = {x ∈ X : x = } is an extreme point of the unit ball BX = {x ∈ X : x ≤ }. Xn} is said to be a strongly orthogonal set relative to an element xi contained in S in the sense of Birkhoff-James iff n xi < xi +
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