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