I remember I’ve seen a theorem but I’m not sure about remembering it truthfully or not. I cannot find this equivalence in anywhere. That is the reason of my doubt. Is it true or not? If it is true how should I show it via using definitions below? Could someone share exact theorem and proof or its link? If there is a mistake please fix me. Thanks a lot
Let $X$ be a vector space over $\mathcal F$ field (it is real or complex numbers) and $A \subseteq X$
$A$ is absolutely convex if and only if A is convex and balanced
Let $\lambda , \beta \in \mathcal F$ and $x,y \in A$
If $|\lambda| + |\beta| \le 1$ such that $\lambda x+ \beta y \in A$ then $A$ is an absolutely convex set
If $\lambda x \in A$ such that $|\lambda| \le 1$ then A is balanced set
If $\lambda x + (1- \lambda )y \in A$ such that $|\lambda| \in [0,1] $ then A is convex set