Absorbing Set in Vector Space contains Zero Vector
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Theorem
Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A \subseteq X$ be an absorbing subset of $X$.
Then:
- $\mathbf 0_X \in A$
where $\mathbf 0_X$ denotes the zero vector in $X$.
Proof
From the definition of an absorbing subset, there exists $t \in \R_{>0}$ such that ${\mathbf 0}_X \in t A$.
So $\mathbf 0_X \in A$.
$\blacksquare$