Bounded and closed subsets of $l^{\infty}$.

Question

How do we define whether a subset of $l^{\infty }$ is bounded and/or closed? I am trying to prove that $B=\{(x_n):||(x_n)||_{l^{\infty}}<=1\}$ is both closed and bounded. It seems to me that boundedness is trivial, by the construction of B. But I am not sure, since I don’t know exactly what bounded and closed mean for B. Thanks in advance!

Answer 1

In any Banach space, a set $S$ is bounded if there exists $M$ such that $\|x\| \le M$ for all $x \in S$.

"Closed" refers to the topology given by the norm. Thus $S$ is closed if its complement is open, i.e. for every $x \notin S$ there exists $\epsilon > 0$ such that the ball $\{y: \|y - x\| < \epsilon\}$ is disjoint from $S$.