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In [[mathematics]] &mdash; specifically, in [[integral|integration theory]] &mdash; the '''Alexiewicz norm''' is an integral [[norm (mathematics)|norm]] associated to the [[Henstock&ndash;Kurzweil integral]].  The Alexiewicz norm turns the space of Henstock&ndash;Kurzweil integrable functions into a [[topological vector space]] that is [[barrelled space|barrelled]] but not [[complete space|complete]].  The Alexiewicz norm is named after the [[Poland|Polish]] mathematician [[Andrzej Alexiewicz]], who introduced it in 1948.

==Definition==

Let HK('''R''') denote the space of all functions ''f'':&nbsp;'''R'''&nbsp;→&nbsp;'''R''' that have finite Henstock&ndash;Kurzweil integral.  Define the '''Alexiewicz semi-norm''' of ''f''&nbsp;∈&nbsp;HK('''R''') by
:<math>\| f \| := \sup \left\{ \left| \int_{I} f \right| : I \subseteq \mathbb{R} \text{ is an interval} \right\}.</math>
This defines a [[semi-norm]] on HK('''R''');  if functions that are equal [[Lebesgue measure|Lebesgue]]-[[almost everywhere]] are identified, then this procedure defines a ''bona fide'' norm on the [[Quotient space (topology)|quotient]] of HK('''R''') by the [[equivalence relation]] of equality almost everywhere.  (Note that the only constant function ''f'':&nbsp;'''R'''&nbsp;→&nbsp;'''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 [[norm (mathematics)#Properties|equivalent]] to the norm defined by
::<math>\| f \|' := \sup_{x \in \mathbb{R}} \left| \int_{- \infty}^{x} f \right|.</math>
* The [[complete metric space|completion]] of HK('''R''') with respect to the Alexiewicz norm is often denoted A('''R''') and is a subspace of the space of [[Distribution (mathematics)|tempered distribution]]s, the dual of [[Schwartz space]].  More precisely, A('''R''') consists of those tempered distributions that are [[distributional derivative]]s of functions in the collection
::<math>\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\}.</math>
:Therefore, if ''f''&nbsp;&isin;&nbsp;A('''R'''), then ''f'' is a tempered distribution and there exists a continuous function ''F'' in the above collection such that
::<math>\langle F', \varphi \rangle = - \langle F, \varphi' \rangle = - \int_{- \infty}^{+ \infty} F \varphi' = \langle f, \varphi \rangle </math>
:for every [[compactly supported]] [[smooth function|''C''<sup>&infin;</sup>]] [[test function]] ''&phi;'':&nbsp;'''R'''&nbsp;&rarr;&nbsp;'''R'''.  In this case, it holds that
::<math>\| f \|' = \sup_{x \in \mathbb{R}} |F(x)| = \| F \|_{\infty}.</math>
* The translation operator is continuous with respect to the Alexiewicz norm.  That is, if for ''f''&nbsp;∈&nbsp;HK('''R''') and ''x''&nbsp;∈&nbsp;'''R''' the translation ''T''<sub>''x''</sub>''f'' of ''f'' by ''x'' is defined by
::<math>(T_{x} f)(y) := f(y - x),</math>
:then
::<math>\| T_{x} f - f \| \to 0 \text{ as } x \to 0.</math>

==References==

* {{cite journal
| last = Alexiewicz
| first = Andrzej
| authorlink = Andrzej Alexiewicz
| title = Linear functionals on Denjoy-integrable functions
| journal = Colloquium Math.
| volume = 1
| year = 1948
| issue = 4
| pages = 289&ndash;293 | doi = 10.4064/cm-1-4-289-293
| mr = 0030120| doi-access = free
}}
* {{cite journal
| last = Talvila
| first = Erik
| title = Continuity in the Alexiewicz norm
| journal = Math. Bohem.
| volume = 131
| year = 2006
| issue = 2
| pages = 189&ndash;196
| doi = 10.21136/MB.2006.134092
| issn = 0862-7959
| url=http://dml.cz/dmlcz/134092 | mr = 2242844| s2cid = 56031790
| doi-access = free
}}

[[Category:Norms (mathematics)]]