FoldUnfold Table of Contents The Minkowski (Gauge) Functional The Minkowski (Gauge) Functional Recall from the Absorbent Sets page that if $X$ is a linear space and $E \subseteq X$ then $E$ is said to be absorbent if for every $x \in X$ there exists a $\lambda > 0$ such that $\lambda x \in E$. Given an absorbent subset $U$ of a linear space $X$ we can define a special type … [Read more...]

## Absorbent Sets

FoldUnfold Table of Contents Absorbent Sets Absorbent Sets Definition: Let $X$ be a linear space and let $E \subseteq X$. Then $E$ is said to be Absorbent or an Absorbing Set if for every $x \in X$ there exists a $\lambda > 0$ such that $\lambda x \in E$. Proposition 1: Let $X$ be a seminormed linear space. Then the open unit ball $B(0, 1) = \{ x \in X : p(x) and the … [Read more...]

## Absolutely Convex Sets

FoldUnfold Table of Contents Absolutely Convex Sets Example 1 Absolutely Convex Sets Definition: Let $X$ be a linear space and let $E \subseteq X$. Then $E$ is said to be Absolutely Convex if whenever $x, y \in E$ and $a, b \in \mathbf{F}$ are such that $|a| + |b| \leq 1$ then $(ax + by) \in E$. Proposition 1: Let $X$ be a linear space and let $E \subseteq X$. If $E$ is … [Read more...]

## Amenability of Abelian Groups

FoldUnfold Table of Contents Amenability of Abelian Groups Amenability of Abelian Groups Theorem 1: Let $G$ be an abelian group. Then $G$ is an amenable group. Recall that a group is said to be abelian (or commutative) if for all $g, h \in G$ we have that $gh = hg$. Proof: Let $A = \ell^1(G)$ and let $\sigma : A \to \mathbb{C}$ be defined for all $a \in A$ … [Read more...]

## ℓ^1(G) is Amenable if and only if G is an Amenable Group 2

FoldUnfold Table of Contents ℓ^1(G) is Amenable if and only if G is an Amenable Group 2 ℓ^1(G) is Amenable if and only if G is an Amenable Group 2 Theorem 1: Let $G$ be a group. Then the group algebra $\ell^1(G)$ is amenable if and only if the group $G$ is an amenable group. $\Leftarrow$ Let $G$ be an amenable group. Then there exists a positive invariant mean $\mu$ on $G$, … [Read more...]