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 HenstockKurzweil 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 HenstockKurzweil integral. Define the Alexiewicz semi-norm of f  HK(R) by

\| 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

\| f \|' := \sup_{x \in \mathbb{R}} \left| \int_{- \infty}^{x} f \right|.
\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
\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
\| f \|' = \sup_{x \in \mathbb{R}} |F(x)| = \| F \|_{\infty}.
(T_{x} f)(y) := f(y - x),
then
\| T_{x} f - f \| \to 0 \text{ as } x \to 0.

References

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