In
mathematics, an Orlicz sequence space is any of certain class of
linear spaces of scalar-valued
sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
, endowed with a special
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
, specified below, under which it forms a
Banach space. Orlicz sequence spaces generalize the
spaces, and as such play an important role in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
.
Definition
Fix
so that
denotes either the real or complex scalar field. We say that a function
is an Orlicz function if it is continuous, nondecreasing, and (perhaps nonstrictly) convex, with
and
. In the special case where there exists
with
for all