Unlike the real numbers, the set of complex numbers $\mathbb C$ cannot be ordered. Therefore, only some concepts applicable to real numbers $\mathbb R$ can be applied to complex numbers as an extension of real numbers. One of these concepts are bounded and unbounded sets.

# Definition: Bounded Complex Sets

A region $U\subseteq\mathbb C$ of complex plane is called:

• bounded, if there exists a real number $C > 0$ such that the absolute value of all points in $U$ does not exceed $C$, formally $$|z|\le C\quad\forall z\in U.$$
• Otherwise, $U$ is called unbounded.

### Examples:

• The second quadrant of the complex plane, i.e. the set $U=\{z=a+bi\in\mathbb C\mid a\ge 0, b\le 0\}$ is unbounded.
• The following region is bounded: Definitions: 1
Propositions: 2

Github: ### References

#### Bibliography

1. Riesz F., Sz.-Nagy, B.: "Functional Analysis (Translation)", Dover Publications, 1955
2. Kneser, Hellmuth: "Mathematische Lehrbücher - Funktionentheorie", Vanderhoeck & Ruprecht, 1958
3. Lang, Serge: "Complex Analysis", Springer, 1999, Forth Edition