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. Orlicz sequence spaces are particular examples of Orlicz spaces.

Definition
Fix $$\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$$ so that $$\mathbb{K}$$ 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 $\lim_{t\to\infty}M(t)=\infty$. In the special case where there exists $$b>0$$ with $$M(t)=0$$ for all $$t\in[0,b]$$ 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 $$(a_n)_{n=1}^\infty\in\mathbb{K}^\mathbb{N}$$ set
 * $$\left\|(a_n)_{n=1}^\infty\right\|_M=\inf\left\{\rho>0:\sum_{n=1}^\infty M(|a_n|/\rho)\leqslant 1\right\}.$$

We then define the Orlicz sequence space with respect to $$M$$, denoted $$\ell_M$$, as the linear space of all $$(a_n)_{n=1}^\infty\in\mathbb{K}^\mathbb{N}$$ such that $\sum_{n=1}^\infty M(|a_n|/\rho)<\infty$ 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
 * $$\limsup_{t\to 0}\frac{M(2t)}{M(t)}<\infty.$$

We denote by $$h_M$$ the subspace of scalar sequences $$(a_n)_{n=1}^\infty\in\ell_M$$ such that $\sum_{n=1}^\infty M(|a_n|/\rho)<\infty$ 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: 1. a_n

2. )<\infty$.

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 $$t\in[0,b]$$. 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
 * $$\liminf_{t\to 0}\frac{tM'(t)}{M(t)}>1\;\;$$ and $$\;\;\limsup_{t\to 0}\frac{tM'(t)}{M(t)}<\infty$$.

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 $\liminf_{t\to 0}tM'(t)/M(t)=\limsup_{t\to 0}tM'(t)/M(t)$ (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 $$t^p$$.