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

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

For each scalar sequence ${\displaystyle (a_{n})_{n=1}^{\infty }\in \mathbb {K} ^{\mathbb {N} }}$  set

${\displaystyle \left\|(a_{n})_{n=1}^{\infty }\right\|_{M}=\inf \left\{\rho >0:\sum _{n=1}^{\infty }M(|a_{n}|/\rho )\leqslant 1\right\}.}$ 

We then define the Orlicz sequence space with respect to ${\displaystyle M}$ , denoted ${\displaystyle \ell _{M}}$ , as the linear space of all ${\displaystyle (a_{n})_{n=1}^{\infty }\in \mathbb {K} ^{\mathbb {N} }}$  such that ${\textstyle \sum _{n=1}^{\infty }M(|a_{n}|/\rho )<\infty }$  for some ${\displaystyle \rho >0}$ , endowed with the norm ${\displaystyle \|\cdot \|_{M}}$ .

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

${\displaystyle \limsup _{t\to 0}{\frac {M(2t)}{M(t)}}<\infty .}$ 

We denote by ${\displaystyle h_{M}}$  the subspace of scalar sequences ${\displaystyle (a_{n})_{n=1}^{\infty }\in \ell _{M}}$  such that ${\textstyle \sum _{n=1}^{\infty }M(|a_{n}|/\rho )<\infty }$  for all ${\displaystyle \rho >0}$ .

The space ${\displaystyle \ell _{M}}$  is a Banach space, and it generalizes the classical ${\displaystyle \ell _{p}}$  spaces in the following precise sense: when ${\displaystyle M(t)=t^{p}}$ , ${\displaystyle 1\leqslant p<\infty }$ , then ${\displaystyle \|\cdot \|_{M}}$  coincides with the ${\displaystyle \ell _{p}}$ -norm, and hence ${\displaystyle \ell _{M}=\ell _{p}}$ ; if ${\displaystyle M}$  is the degenerate Orlicz function then ${\displaystyle \|\cdot \|_{M}}$  coincides with the ${\displaystyle \ell _{\infty }}$ -norm, and hence ${\displaystyle \ell _{M}=\ell _{\infty }}$  in this special case, and ${\displaystyle h_{M}=c_{0}}$  when ${\displaystyle M}$  is degenerate.

In general, the unit vectors may not form a basis for ${\displaystyle \ell _{M}}$ , and hence the following result is of considerable importance.

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

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

${\displaystyle \ell _{M}}$  can be isomorphic to ${\displaystyle \ell _{N}}$  without their unit vector bases being equivalent. (See the example below of an Orlicz sequence space with two nonequivalent symmetric bases.)

Theorem 2. Let ${\displaystyle M}$  be an Orlicz function. Then ${\displaystyle \ell _{M}}$  is reflexive if and only if

${\displaystyle \liminf _{t\to 0}{\frac {tM'(t)}{M(t)}}>1\;\;}$  and ${\displaystyle \;\;\limsup _{t\to 0}{\frac {tM'(t)}{M(t)}}<\infty }$ .

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

Theorem 4 (Lindenstrauss/Tzafriri). Every separable Orlicz sequence space ${\displaystyle \ell _{M}}$  contains a subspace isomorphic to ${\displaystyle \ell _{p}}$  for some ${\displaystyle 1\leqslant p<\infty }$ .

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

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

Theorem 5 (K. J. Lindberg & Lindenstrauss/Tzafriri). If ${\displaystyle \ell _{M}}$  is an Orlicz sequence space satisfying ${\textstyle \liminf _{t\to 0}tM'(t)/M(t)=\limsup _{t\to 0}tM'(t)/M(t)}$  (i.e., the two-sided limit exists) then the following are all true.

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

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