◀ ▲ ▶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

• This is an example of how bounded subsets can be defined for an unordered set.

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:

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Definitions: 1
Propositions: 2

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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