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 [math]\displaystyle{ \ell_p }[/math] spaces, and as such play an important role in functional analysis. Orlicz sequence spaces are particular examples of Orlicz spaces.

Definition

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

In what follows, unless otherwise stated we'll assume all Orlicz functions are nondegenerate. This implies [math]\displaystyle{ M(t)\gt 0 }[/math] for all [math]\displaystyle{ t\gt 0 }[/math].

For each scalar sequence [math]\displaystyle{ (a_n)_{n=1}^\infty\in\mathbb{K}^\mathbb{N} }[/math] set

[math]\displaystyle{ \left\|(a_n)_{n=1}^\infty\right\|_M=\inf\left\{\rho\gt 0:\sum_{n=1}^\infty M(|a_n|/\rho)\leqslant 1\right\}. }[/math]

We then define the Orlicz sequence space with respect to [math]\displaystyle{ M }[/math], denoted [math]\displaystyle{ \ell_M }[/math], as the linear space of all [math]\displaystyle{ (a_n)_{n=1}^\infty\in\mathbb{K}^\mathbb{N} }[/math] such that [math]\displaystyle{ \sum_{n=1}^\infty M(|a_n|/\rho)\lt \infty }[/math] for some [math]\displaystyle{ \rho\gt 0 }[/math], endowed with the norm [math]\displaystyle{ \|\cdot\|_M }[/math].

Two other definitions will be important in the ensuing discussion. An Orlicz function [math]\displaystyle{ M }[/math] is said to satisfy the Δ₂ condition at zero whenever

[math]\displaystyle{ \limsup_{t\to 0}\frac{M(2t)}{M(t)}\lt \infty. }[/math]

We denote by [math]\displaystyle{ h_M }[/math] the subspace of scalar sequences [math]\displaystyle{ (a_n)_{n=1}^\infty\in\ell_M }[/math] such that [math]\displaystyle{ \sum_{n=1}^\infty M(|a_n|/\rho)\lt \infty }[/math] for all [math]\displaystyle{ \rho\gt 0 }[/math].

Properties

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

In general, the unit vectors may not form a basis for [math]\displaystyle{ \ell_M }[/math], and hence the following result is of considerable importance.

Theorem 1. If [math]\displaystyle{ M }[/math] is an Orlicz function then the following conditions are equivalent:

1. [math]\displaystyle{ M }[/math] satisfies the Δ₂ condition at zero, i.e. [math]\displaystyle{ \limsup_{t\to 0}M(2t)/M(t)\lt \infty }[/math].
2. For every [math]\displaystyle{ \lambda\gt 0 }[/math] there exists positive constants [math]\displaystyle{ K=K(\lambda) }[/math] and [math]\displaystyle{ b=b(\lambda) }[/math] so that [math]\displaystyle{ M(\lambda t)\leqslant KM(t) }[/math] for all [math]\displaystyle{ t\in[0,b] }[/math].
3. [math]\displaystyle{ \limsup_{t\to 0}tM'(t)/M(t)\lt \infty }[/math] (where [math]\displaystyle{ M' }[/math] 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).
4. [math]\displaystyle{ \ell_M=h_M }[/math].
5. The unit vectors form a boundedly complete symmetric basis for [math]\displaystyle{ \ell_M }[/math].
6. [math]\displaystyle{ \ell_M }[/math] is separable.
7. [math]\displaystyle{ \ell_M }[/math] fails to contain any subspace isomorphic to [math]\displaystyle{ \ell_\infty }[/math].
8. [math]\displaystyle{ (a_n)_{n=1}^\infty\in\ell_M }[/math] if and only if [math]\displaystyle{ \sum_{n=1}^\infty M(|a_n|)\lt \infty }[/math].

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

[math]\displaystyle{ \ell_M }[/math] can be isomorphic to [math]\displaystyle{ \ell_N }[/math] without their unit vector bases being equivalent. (See the example below of an Orlicz sequence space with two nonequivalent symmetric bases.)

Theorem 2. Let [math]\displaystyle{ M }[/math] be an Orlicz function. Then [math]\displaystyle{ \ell_M }[/math] is reflexive if and only if

[math]\displaystyle{ \liminf_{t\to 0}\frac{tM'(t)}{M(t)}\gt 1\;\; }[/math] and [math]\displaystyle{ \;\;\limsup_{t\to 0}\frac{tM'(t)}{M(t)}\lt \infty }[/math].

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

Theorem 4 (Lindenstrauss/Tzafriri). Every separable Orlicz sequence space [math]\displaystyle{ \ell_M }[/math] contains a subspace isomorphic to [math]\displaystyle{ \ell_p }[/math] for some [math]\displaystyle{ 1\leqslant p\lt \infty }[/math].

Corollary. Every infinite-dimensional closed subspace of a separable Orlicz sequence space contains a further subspace isomorphic to [math]\displaystyle{ \ell_p }[/math] for some [math]\displaystyle{ 1\leqslant p\lt \infty }[/math].

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

Theorem 5 (K. J. Lindberg & Lindenstrauss/Tzafriri). If [math]\displaystyle{ \ell_M }[/math] is an Orlicz sequence space satisfying [math]\displaystyle{ \liminf_{t\to 0}tM'(t)/M(t)=\limsup_{t\to 0}tM'(t)/M(t) }[/math] (i.e., the two-sided limit exists) then the following are all true.

1. [math]\displaystyle{ \ell_M }[/math] is separable.
2. [math]\displaystyle{ \ell_M }[/math] contains a complemented copy of [math]\displaystyle{ \ell_p }[/math] for some [math]\displaystyle{ 1\leqslant p\lt \infty }[/math].
3. [math]\displaystyle{ \ell_M }[/math] has a unique symmetric basis (up to equivalence).

Example. For each [math]\displaystyle{ 1\leqslant p\lt \infty }[/math], the Orlicz function [math]\displaystyle{ M(t)=t^p/(1-\log (t)) }[/math] satisfies the conditions of Theorem 5 above, but is not equivalent to [math]\displaystyle{ t^p }[/math].

References

• Classical Banach Spaces I, Sequence Spaces, 1977, ISBN 978-3-642-66559-2 
• "On Orlicz Sequence Spaces". Israel Journal of Mathematics 10 (3): 379–390. September 1971. doi:10.1007/BF02771656. 
• "On Orlicz Sequence Spaces. II". Israel Journal of Mathematics 11 (4): 355–379. December 1972. doi:10.1007/BF02761463. 
• "On Orlicz Sequence Spaces III". Israel Journal of Mathematics 14 (4): 368–389. December 1973. doi:10.1007/BF02764715.