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(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

${\displaystyle \|f\|:=\sup \left\{\left|\int _{I}f\right|:I\subseteq \mathbb {R} {\text{ is an interval}}\right\}.}$

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[]

• 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

${\displaystyle \|f\|':=\sup _{x\in \mathbb {R} }\left|\int _{-\infty }^{x}f\right|.}$

• 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

${\displaystyle \left\{F\colon \mathbb {R} \to \mathbb {R} \,\left|\,F{\text{ is continuous, }}\lim _{x\to -\infty }F(x)=0,\lim _{x\to +\infty }F(x)\in \mathbb {R} \right.\right\}.}$

Therefore, if f ∈ A(R), then f is a tempered distribution and there exists a continuous function F in the above collection such that

${\displaystyle \langle F',\varphi \rangle =-\langle F,\varphi '\rangle =-\int _{-\infty }^{+\infty }F\varphi '=\langle f,\varphi \rangle }$

for every compactly supported C^∞ test function φ: R → R. In this case, it holds that

${\displaystyle \|f\|'=\sup _{x\in \mathbb {R} }|F(x)|=\|F\|_{\infty }.}$

• The translation operator is continuous with respect to the Alexiewicz norm. That is, if for f ∈ HK(R) and x ∈ R the translation T_xf of f by x is defined by

${\displaystyle (T_{x}f)(y):=f(y-x),}$

then

${\displaystyle \|T_{x}f-f\|\to 0{\text{ as }}x\to 0.}$

References[]

• Alexiewicz, Andrzej (1948). "Linear functionals on Denjoy-integrable functions". Colloquium Math. 1: 289–293. MR 0030120.
• Talvila, Erik (2006). "Continuity in the Alexiewicz norm". Math. Bohem. 131 (2): 189–196. ISSN 0862-7959. MR 2242844.