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[edit]
Let HK(R) denote the space of all functions f: R → R that have finite Henstock–Kurzweil integral. Define the Alexiewicz semi-norm of f ∈ HK(R) by
This defines a semi-norm on HK(R); if functions that are equal Lebesgue-almost everywhere are identified, then this procedure defines a bona fide norm on the quotient of HK(R) by the equivalence relation of equality almost everywhere. (Note that the only constant function f: R → R that is integrable is the one with constant value zero.)
Properties[edit]
- The Alexiewicz norm endows HK(R) with a topology that is barrelled but incomplete.
- The Alexiewicz norm as defined above is equivalent to the norm defined by
- The completion of HK(R) with respect to the Alexiewicz norm is often denoted A(R) and is a subspace of the space of tempered distributions, the dual of Schwartz space. More precisely, A(R) consists of those tempered distributions that are distributional derivatives of functions in the collection
- Therefore, if f ∈ A(R), then f is a tempered distribution and there exists a continuous function F in the above collection such that
- for every compactly supported C∞ test function φ: R → R. In this case, it holds that
- The translation operator is continuous with respect to the Alexiewicz norm. That is, if for f ∈ HK(R) and x ∈ R the translation Txf of f by x is defined by
- then
References[edit]
- Alexiewicz, Andrzej (1948). "Linear functionals on Denjoy-integrable functions". Colloquium Math. 1 (4): 289–293. doi:10.4064/cm-1-4-289-293. MR 0030120.
- Talvila, Erik (2006). "Continuity in the Alexiewicz norm". Math. Bohem. 131 (2): 189–196. doi:10.21136/MB.2006.134092. ISSN 0862-7959. MR 2242844. S2CID 56031790.