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

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
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 φR → 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]