Definition:Bounded Mapping/Complex-Valued

Definition

Let $f: S \to \C$ be a complex-valued function.

Then $f$ is bounded if and only if the real-valued function $\cmod f: S \to \R$ is bounded, where $\cmod f$ is the modulus of $f$.

That is, $f$ is bounded if there is a constant $K \ge 0$ such that $\cmod {\map f z} \le K$ for all $z \in S$.

Unbounded

Let $f: S \to \C$ be a complex-valued function.

Then $f$ is unbounded if and only if $f$ is not bounded.

That is, $f$ is unbounded if there does not exist a constant $K \ge 0$ such that $\cmod {f \paren z} \le K$ for all $z \in S$.

Also see

• Complex Plane is Metric Space: this definition coincides with the definition of a bounded mapping to a metric space, using the standard metric on $\C$.

• Results about bounded complex-valued functions can be found here.

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.