# Definition:Bounded Sequence/Normed Vector Space

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*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