Properties of Absorbent Sets of Vectors

Recall that if $E$ is a vector space then a subset $A$ of $E$ is absorbent if for each $x \in E$ there exists a $\lambda > 0$ such that if $\mu \in \mathbf{F}$ is such that $|\mu| \geq \lambda$ then $x \in \mu A$. We will now look at some useful results regarding absorbent sets.

Proposition 1: Let $E$ be a vector space. If $A \subseteq E$ is absorbent then for each $x \in E$ there exists a $\mu_0$ with $0 < \mu_0 < 1$ such that $-\mu_0 x \in A$.

• Proof: Let $A$ be absorbent. Given $x \in E$, there exists a $\lambda > 0$ such that $x \in \mu A$ for all $\mu \in \mathbf{F}$ with $|\mu| \geq \lambda$. In particular, for each $x \in E$, $- \max \{ \lambda, 2 \} \in \mathbf{F}$ is such that:

(1)

\begin{align} |-\max \{ \lambda, 2 \}| = \max \{ \lambda, 2 \} \geq \lambda \end{align}

• and so $x \in -\max \{ \lambda, 2 \} A$ and $\displaystyle{- \frac{1}{\max \{ \lambda, 2 \}} x \in A}$. By setting $\mu_0 = \frac{1}{\max \{\lambda, 2 \}}$, we see that $0 < \mu_0 < 1$ and is such that $-\mu_0 x \in A$. $\blacksquare$