Definition:Bounded Mapping/Normed Vector Space

Definition

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

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

Let $S$ be a set.

Let $f : S \to X$ be a mapping.

We say that $f$ is bounded if and only if there exists a real number $M > 0$ such that:

$\norm {\map f x} \le M$ for each $x \in S$.

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