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Luxemburg norm

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A function

where is an even convex function that increases for positive ,

for , and is a bounded set in . The properties of this norm were studied by W.A.J. Luxemburg [1]. The Luxemburg norm is equivalent to the Orlicz norm (see Orlicz space), and

If the functions and are complementary (or dual) to each other (see Orlicz class), then

If is the characteristic function of a measurable subset , then

References

[1] W.A.J. Luxemburg, "Banach function spaces" , T.U. Delft (1955) (Thesis)
[2] M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (Translated from Russian)
How to Cite This Entry:
Luxemburg norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luxemburg_norm&oldid=47719
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article