Alexiewicz norm

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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 fR → R that have finite Henstock–Kurzweil 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 fR → 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
\| 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 φR → R. In this case, it holds that
\| 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 Txf of f by x is defined by
(T_{x} f)(y) := f(y - x),
then
\| T_{x} f - f \| \to 0 \text{ as } x \to 0.

References[edit]