Definition: Supremum Norm for Functions

Let $D$ be a set, $\mathbb F$ be a either the field of real numbers or the field of complex numbers and let $f:D\to\mathbb F$ be a function. We set $$||f||_\infty:=\sup\{|f(x)|\mid x\in D\},$$ where $|f(x)|$ is the absolute value of real numbers (or the absolute value of complex numbers) and call $||f||_\infty$ the supremum norm of $f$ on $D.$

Notes

Proofs: 1 2 3 4
Propositions: 5 6 7


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References

Bibliography

  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983