# Bounded set and norm bounded set in a Banach lattice space

More precisely, let $$E$$ be a Banach lattice space with the order $$\le$$. Let $$\left \| . \right \|$$ be the norm on $$E$$.
A subset $$A$$ of $$E$$ is said to be order bounded if there exist $$x, y \in E$$ such that $$x \le a \le y$$ holds for each $$a\in A$$.
Besides, a subset $$A$$ of $$E$$ is said to be a norm bounded set if there exists a non-negative number $$M$$ such that $$\left \| x \right \| \le M$$, for every $$x \in A$$.