Luxemburg norm

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)