Supremum norm

The supremum norm of a function is the supremum of its absolute value over a specified domain. It measures the function’s greatest magnitude, whether or not that value is attained.

Supremum: The least upper bound of a set in an ordered space. It supplies the least upper bound used to define the norm, even when no value is attained.
Uniform norm: A norm measuring the largest pointwise magnitude of a function, often written as a supremum. This is a widely used name for the supremum norm, especially in analysis.
Space of continuous functions: A vector space consisting of continuous functions on a specified domain. On compact domains, continuous functions have finite supremum norm and form a central normed space.
L1 norm: A norm measuring the integral or sum of absolute values. It measures aggregate magnitude rather than the single largest pointwise value.
Maximum: The greatest element of a set, which belongs to the set. A supremum norm can exist even when the function has no maximum magnitude.
L-infinity norm: The norm associated with the essential supremum of a measurable function. It is the measure-theoretic counterpart, ignoring changes on sets of measure zero.
Banach space: A normed vector space that is complete under the distance induced by its norm. The supremum norm makes bounded continuous functions on suitable domains a standard complete space.
L2 norm: A norm derived from the square root of the integral or sum of squared magnitudes. It emphasizes average quadratic size instead of worst-case deviation.
Bounded function: A function whose values remain within a finite bound on its domain. Boundedness makes the supremum norm finite.
Essential supremum: The smallest bound that a measurable function satisfies almost everywhere. It replaces the ordinary supremum when functions equal almost everywhere are identified.

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