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PhyslibAlpha.Mathematics.Sublinear

Linear minorants of sublinear functionals

i. Overview

A sublinear functional is the largest of the linear functionals below it. At every point the Hahn–Banach theorem gives a linear functional below it that agrees with it there.

ii. Key results

- `exists_linearMap_le_eq_of_sublinear` proves that a linear functional below a sublinear functional can be chosen to agree with it at a given point.

iii. Table of contents

  • A. Linear minorants

A. Linear minorants

3 declarations

theorem

N(0)=0N(0) = 0 for sublinear functionals

For any sublinear functional NN, the value of the functional at the zero vector is zero, that is, N(0)=0N(0) = 0.

theorem

cN(x)≤N(c⋅x)c N(x) \le N(c \cdot x) for sublinear functionals

Let VV be a real vector space and N:V→RN: V \to \mathbb{R} be a sublinear functional. For any scalar c∈Rc \in \mathbb{R} and any vector x∈Vx \in V, the following inequality holds: cN(x)≤N(c⋅x)c N(x) \le N(c \cdot x) In other words, a sublinear functional dominates the multiples of its value at any point.

theorem

Existence of a linear minorant attaining a sublinear functional at xx

Let VV be a real vector space and N:V→RN: V \to \mathbb{R} be a sublinear functional. For any point x∈Vx \in V, there exists a linear functional g:V→Rg: V \to \mathbb{R} such that g(y)≤N(y)g(y) \leq N(y) for all y∈Vy \in V, and g(x)=N(x)g(x) = N(x). This is a version of the Hahn–Banach theorem stating that a sublinear functional has a linear minorant that attains its value at a given point xx.