Hi I am learning functional analysis via the lectures notes of someone I know. His professor has a less common definition of "Balanced Set", so I am checking my understanding:
A neighborhood $X$ of $0$ is said to be balanced if for every $x\in X$ and $r \in [0,1]$, we have $rx \in X$.
Using the cartesian plane $\mathbb{R}^2$ as my intuition, I would interpret a balanced neighborhood $X$ of $0$ to be any neighborhood that contains whole line segments connecting $(0,0)$ to points $(x,y) \in X$.
I am ok so far. But then the lectures state this, which looks like the usual definition:
This is equivalent to saying that $X$ is balanced if for every $x\in X$ and $|r| \leq 1$, we have $rX \subseteq X$
The first definition seems like a so called "Star Domain", and the 2 don't seem equivalent. Right?