Definition:Bounded/Metric Space
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Definition
Metric Space
Let $M = \left({X, d}\right)$ be a metric space.
From the definition of a metric, $d: X \times X \to \R$ is a real-valued function.
Hence we can define that a metric space $\left({X, d}\right)$ is bounded if there exists $a \in X$ and $K \in \R$ such that $d \left({x, a}\right) \le K$ for all $x \in S$.
It follows immediately that, if $M$ satisfies this condition for one $a \in X$, then it does so for all $a' \in X$, with $K$ replaced by $K^{\prime} = K + d \left({a, a^{\prime}}\right)$.
This is because $d \left({x, a}\right) \le K \implies d \left({x, a^{\prime}}\right) \le d \left({x, a}\right) + d \left({a, a^{\prime}}\right) \le K + d \left({a, a^{\prime}}\right)$.
Metric Subspace
Let $M = \left({X, d}\right)$ be a metric space.
Let $M' = \left({Y, d_Y}\right)$ be a subspace of $M$.
Then $M'$ is bounded (in $M$) if $M'$ is bounded with respect to the subspace metric $d_Y$.
Mapping into Metric Space
Let $M$ be a metric space.
Let $f: X \to M$ be a mapping from any set $X$ into $M$.
Then $f$ is called bounded if $f \left({X}\right)$ is bounded in $M$.
Mapping into Real Number Line
Note that as the real numbers form a metric space, we can in theory consider defining boundedness on a real-valued function in terms of boundedness of a mapping into a metric space.
However, as a metric space is itself defined in terms of a real-valued function in the first place, this concept can be criticised as being a circular definition.
Unbounded
Any space which is not bounded is described as unbounded.
Also see
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$: Complete Metric Spaces
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definitions $2.2.12, \ 2.2.14$
