Answer to Question #275549 in Functional Analysis for Himanshu

Question

Show that a norm on a vector space X is a sublinear functional on X

Answer 1

Let’s 𝑓(𝑥) = ||𝑥||. Obviously, f(x) is a function from a vector space 𝑋 to the scalar field ℝ. 1. ∀𝑥 ∈ 𝑋, ∀𝑎 ∈ ℝ+, 𝑓(𝑥) = ||𝑎 ∙ 𝑥|| = |𝑎| ∙ ||𝑥|| = 𝑎 ∙ ||𝑥|| = 𝑎 ∙ 𝑓(𝑥), due to the multiplicative property of a norm. 2. ∀𝑥, 𝑦 ∈ 𝑋, 𝑓(𝑥 + 𝑦) = ||𝑥 + 𝑦|| ≤ ||𝑥|| + ||𝑦|| = 𝑓(𝑥) + 𝑓(𝑦), because of the triangle inequality. Correctness of the statement and both properties (positive homogeneity and subadditivity) were proved.