Definition:Bounded

Ordered Set
Let $$\left({S; \preceq}\right)$$ be a poset.

Let $$T \subseteq S$$ be both bounded below and bounded above in $$S$$.

Then $$T$$ is bounded in $$S$$.

Mapping
Let $$\left({T; \preceq}\right)$$ be a poset.

Let $$f: S \to T$$ be a mapping.

Let the range of $$f$$ be bounded.

Then $$f$$ is defined as being bounded.

That is, $$f$$ is bounded if it is both bounded above and bounded below.

Metric Space
A metric space $$(S,d)$$ is called bounded if there exist $$a\in S$$ and $$K\in\R$$ such that $$d(x,a)\leq K$$ for all $$x\in S$$.

It follows immediately that, if $$S$$ satisfies this condition for one $$a\in S$$, then it does so for all $$a'$$, with $$K$$ replaced by $$K^{\prime} = K + d \left({a, a^{\prime}}\right)$$. This is because $$d \left({x, a}\right) \le K \Longrightarrow d \left({x, a^{\prime}}\right) \le d \left({x, a}\right) + d \left({a, a^{\prime}}\right) \le K + d \left({a, a^{\prime}}\right)$$.

If $$M$$ is a metric space and $$S$$ is a subset of M, then we say that $$S$$ is bounded (in $$M$$) if $$S$$ is bounded with respect to the subspace metric.

Mapping into Metric Space
Let $$M = \left\{{A, d}\right\}$$ be a metric space, and suppose that $$f: X \to M$$ is a mapping from any set $$X$$ into $$M$$.

Then $$f$$ is called bounded if $$f \left({X}\right)$$ is bounded in $M$.

Totally Bounded Metric Space
A metric space $$(S,d)$$ is called totally bounded if, for every $$\varepsilon>0$$, there exist finitely many points $$x_0,\dots,x_n\in S$$ such that
 * $$\sup_{0\leq i \leq n} d(x_i,x)\leq \varepsilon$$

for all $$x\in S$$.

Any totally bounded metric space is also bounded (pick $$x_0,\dots,x_n$$ for $$\varepsilon=1$$ as in the definition of total boundedness, and use $$a=x_0$$ and $$K=2(n+1)$$ in the definition of boundedness). The converse is not true. However, a subset $$S$$ of a complete metric space $$M$$ is totally bounded if and only if it is bounded; hence the two concepts coincide in most standard examples.

A metric space is compact if and only if it is complete and totally bounded.

Real-valued Function
A real-valued function $$f:S\to\R$$ is bounded if there is a number $$K\geq 0$$ such that $$|f(x)|\leq K$$ for all $$x\in S$$.

Note that this coincides with the above definitions of bounded functions (using the standard ordering / metric on $$\R$$); see Bounded Set of Real Numbers‎ and Real Number Line is Metric Space.

Complex-Valued Function
A complex-valued function $$f:S\to\C$$ is called bounded if the real-valued function $$|f|:S\to\R$$ is bounded. That is, $$f$$ is bounded if there is a constant $$K\geq 0$$ such that $$|f(z)|\leq K$$ for all $$z\in S$$.

This coincides with the definition of a bounded mapping into a metric space, using the standard metric on $$\C$$; seeComplex Plane is Metric Space.