Definition:Bounded Sequence/Normed Vector Space

This page is about Bounded Sequence in the context of Normed Vector Space. For other uses, see Bounded Sequence.

Definition

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $\sequence {x_n}$ be a sequence in $X$.

Then $\sequence {x_n}$ is bounded if and only if:

$\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$

Unbounded

$\sequence {x_n}$ is unbounded if and only if it is not bounded.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $1$: Normed and Banach spaces