◀ ▲ ▶Branches / Analysis / Definition: Bounded Complex Sets
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.
Notes
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:
Mentioned in:
Definitions: 1
Propositions: 2
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References
Bibliography
 Riesz F., Sz.Nagy, B.: "Functional Analysis (Translation)", Dover Publications, 1955
 Kneser, Hellmuth: "Mathematische Lehrbücher  Funktionentheorie", Vanderhoeck & Ruprecht, 1958
 Lang, Serge: "Complex Analysis", Springer, 1999, Forth Edition