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 $D.$

$||f||_\infty$ is either a non-negative real number or $+\infty,$ because $f$ can be an arbitrary function.
$||f||_\infty\neq+\infty$ if and only if $|f|$ is bounded on $D.$
The supremum norm is a generalization of the maximum norm which was defined for vectors. For functions, the maximum value of $|f(x)|$ for all $x\in D$ does not have to exist (i.e. be an element of the image of the absolute value of the function $|f(D)|$). Therefore, the supremum replaces the maximum used in the original definition.
Note that the supremum norm works fine even in the case if $D$ has infinitely many (even uncountably infinitely many) elements. This was not the case for the maximum norm.

Proofs: 1 2 3 4
Propositions: 5 6 7

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