# Orlicz sequence space

In mathematics, an **Orlicz sequence space** is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, 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.

## 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
it is called **degenerate**.

In what follows, unless otherwise stated we'll assume all Orlicz functions are nondegenerate. Note that this implies for all .

For each scalar sequence set

We then define the **Orlicz sequence space** with respect to
, denoted
, as the linear space of all
such that
for some
, endowed with the norm
.

Two other definitions will be important in the ensuing discussion. An Orlicz function
is said to satisfy the **Δ _{2} condition at zero** whenever

We denote by the subspace of scalar sequences such that for all .

## Properties

The space is a Banach space, and it generalizes the classical spaces in the following precise sense: when , , then coincides with the -norm, and hence ; if is the degenerate Orlicz function then coincides with the -norm, and hence in this special case. Note also that when is degenerate.

In general, the unit vectors may not form a basis for , and hence the following result is of considerable importance.

**Theorem 1.** If
is an Orlicz function then the following conditions are equivalent:

- (i)
satisfies the Δ
_{2}condition at zero, i.e. .

- (ii) For every there exists positive constants and so that for all .

- (iii) (where is a nondecreasing function defined everywhere except perhaps on a countable set, where instead we can take the right-hand derivative which is defined everywhere).

- (iv) .

- (v) The unit vectors form a boundedly complete symmetric basis for .

- (vi) is separable.

- (vii) fails to contain any subspace isomorphic to .

- (viii) if and only if .

Two Orlicz functions
and
satisfying the Δ_{2} condition at zero are called **equivalent** whenever there exist are positive constants
such that
for all
. This is the case if and only if the unit vector bases of
and
are equivalent.

Note that can be isomorphic to without their unit vector bases being equivalent. (See the example below of an Orlicz sequence space with two nonequivalent symmetric bases.)

**Theorem 2.** Let
be an Orlicz function. Then
is reflexive if and only if

- and .

**Theorem 3** (K. J. Lindberg). Let
be an infinite-dimensional closed subspace of a separable Orlicz sequence space
. Then
has a subspace
isomorphic to some Orlicz sequence space
for some Orlicz function
satisfying the Δ_{2} condition at zero. If furthermore
has an unconditional basis then
may be chosen to be complemented in
, and if
has a symmetric basis then
itself is isomorphic to
.

**Theorem 4** (Lindenstrauss/Tzafriri). Every separable Orlicz sequence space
contains a subspace isomorphic to
for some
.

**Corollary.** Every infinite-dimensional closed subspace of a separable Orlicz sequence space contains a further subspace isomorphic to
for some
.

Note that in the above Theorem 4, the copy of may not always be chosen to be complemented, as the following example shows.

**Example** (Lindenstrauss/Tzafriri). There exists a separable and reflexive Orlicz sequence space
which fails to contain a complemented copy of
for any
. This same space
contains at least two nonequivalent symmetric bases.

**Theorem 5** (K. J. Lindberg & Lindenstrauss/Tzafriri). If
is an Orlicz sequence space satisfying
(i.e., the two-sided limit exists) then the following are all true.

- (i) is separable.

- (ii) contains a complemented copy of for some .

- (iii) has a unique symmetric basis (up to equivalence).

**Example.** For each
, the Orlicz function
satisfies the conditions of Theorem 5 above, but is not equivalent to
.

## References

- Lindenstrauss, J., and L. Tzafriri.
*Classical Banach Spaces I, Sequence Spaces*(1977), ISBN 978-3-642-66559-2. - Lindenstrass, J., and L. Tzafriri. "On Orlicz Sequence Spaces,"
*Israel Journal of Mathematics*10:3 (Sep 1971), pp379-390. - Lindenstrass, J., and L. Tzafriri. "On Orlicz sequence spaces. II,"
*Israel Journal of Mathematics*11:4 (Dec 1972), pp355-379. - Lindenstrass, J., and L. Tzafriri. "On Orlicz sequence spaces III,"
*Israel Journal of Mathematics*14:4 (Dec 1973), pp368-389.