Alexiewicz norm


In mathematics — specifically, in integration theory — the Alexiewicz norm is an integral norm associated to the Henstock-Kurzweil integral. The Alexiewicz norm turns the space of Henstock-Kurzweil integrable functions into a topological vector space that is barrelled but not complete. The Alexiewicz norm is named after the Polish mathematician Andrzej Alexiewicz, who introduced it in 1948.

Definition

Let HK denote the space of all functions f: R → R that have finite Henstock-Kurzweil integral. Define the Alexiewicz semi-norm of f ∈ HK by
This defines a semi-norm on HK; if functions that are equal Lebesgue-almost everywhere are identified, then this procedure defines a bona fide norm on the quotient of HK by the equivalence relation of equality almost everywhere.

Properties