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 $\ell_p$ spaces, and as such play an important role in functional analysis.

Definition

Fix $\mathbb\in\$ so that $\mathbb$ denotes either the real or complex scalar field. We say that a function $M:[0,\infty)\to[0,\infty)$ is an Orlicz function if it is continuous, nondecreasing, and (perhaps nonstrictly) convex, with $M(0)=0$ and $\lim_M(t)=\infty$. In the special case where there exists $b>0$ with $M(t)=0$ for all t\in ,b/math> it is called degenerate.

In what follows, unless otherwise stated we'll assume all Orlicz functions are nondegenerate. This implies $M(t)>0$ for all $t>0$.

For each scalar sequence $(a_n)_^\infty\in\mathbb^\mathbb$ set
:$\left\, (a_n)_^\infty\right\, _M=\inf\left\.$
We then define the Orlicz sequence space with respect to $M$, denoted $\ell_M$, as the linear space of all $(a_n)_^\infty\in\mathbb^\mathbb$ such that $\sum_^\infty M(, a_n, /\rho)<\infty$ for some $\rho>0$, endowed with the norm $\, \cdot\, _M$.

Two other definitions will be important in the ensuing discussion. An Orlicz function $M$ is said to satisfy the Δ₂ condition at zero whenever
:$\limsup_\frac<\infty.$
We denote by $h_M$ the subspace of scalar sequences $(a_n)_^\infty\in\ell_M$ such that $\sum_^\infty M(, a_n, /\rho)<\infty$ for all $\rho>0$.

Properties

The space $\ell_M$ is a Banach space, and it generalizes the classical $\ell_p$ spaces in the following precise sense: when $M(t)=t^p$, $1\leqslant p<\infty$, then $\, \cdot\, _M$ coincides with the $\ell_p$-norm, and hence $\ell_M=\ell_p$; if $M$ is the degenerate Orlicz function then $\, \cdot\, _M$ coincides with the $\ell_\infty$-norm, and hence $\ell_M=\ell_\infty$ in this special case, and $h_M=c_0$ when $M$ is degenerate.

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

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

Two Orlicz functions $M$ and $N$ satisfying the Δ₂ condition at zero are called equivalent whenever there exist are positive constants $A,B,b>0$ such that $AN(t)\leqslant M(t)\leqslant BN(t)$ for all t\in ,b/math>. This is the case if and only if the unit vector bases of $\ell_M$ and $\ell_N$ are equivalent.

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

Theorem 2. Let $M$ be an Orlicz function. Then $\ell_M$ is reflexive if and only if
: $\liminf_\frac>1\;\;$ and $\;\;\limsup_\frac<\infty$.

Theorem 3 (K. J. Lindberg). Let $X$ be an infinite-dimensional closed subspace of a separable Orlicz sequence space $\ell_M$. Then $X$ has a subspace $Y$ isomorphic to some Orlicz sequence space $\ell_N$ for some Orlicz function $N$ satisfying the Δ₂ condition at zero. If furthermore $X$ has an unconditional basis then $Y$ may be chosen to be complemented in $X$, and if $X$ has a symmetric basis then $X$ itself is isomorphic to $\ell_N$.

Theorem 4 (Lindenstrauss/Tzafriri). Every separable Orlicz sequence space $\ell_M$ contains a subspace isomorphic to $\ell_p$ for some $1\leqslant p<\infty$.

Corollary. Every infinite-dimensional closed subspace of a separable Orlicz sequence space contains a further subspace isomorphic to $\ell_p$ for some $1\leqslant p<\infty$.

Note that in the above Theorem 4, the copy of $\ell_p$ 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 $\ell_M$ which fails to contain a complemented copy of $\ell_p$ for any $1\leqslant p\leqslant\infty$. This same space $\ell_M$ contains at least two nonequivalent symmetric bases.

Theorem 5 (K. J. Lindberg & Lindenstrauss/Tzafriri). If $\ell_M$ is an Orlicz sequence space satisfying $\liminf_tM'(t)/M(t)=\limsup_tM'(t)/M(t)$ (i.e., the two-sided limit exists) then the following are all true.

Example. For each $1\leqslant p<\infty$, the Orlicz function $M(t)=t^p/(1-\log (t))$ satisfies the conditions of Theorem 5 above, but is not equivalent to $t^p$.

References

*
*
*
*{{cite journal
, last1=Lindenstrauss , first1=Joram , authorlink1=Joram Lindenstrauss
, last2=Tzafriri , first2=Lior
, title=On Orlicz Sequence Spaces III
, journal=Israel Journal of Mathematics
, volume=14
, issue=4
, pages=368–389
, date=December 1973
, doi=10.1007/BF02764715 , doi-access=free

Functional analysis
Sequence spaces