◀ ▲ ▶Branches / Analysis / Definition: Supremum Norm for Functions
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
 $f_\infty$ is either a nonnegative 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.
Mentioned in:
Proofs: 1 2 3 4
Propositions: 5 6 7
Thank you to the contributors under CC BYSA 4.0!
 Github:

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