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 formula_1 spaces, and as such play an important role in functional analysis.
Definition.
Fix formula_2 so that formula_3 denotes either the real or complex scalar field. We say that a function formula_4 is an Orlicz function if it is continuous, nondecreasing, and (perhaps nonstrictly) convex, with formula_5 and formula_6. In the special case where there exists formula_7 with formula_8 for all formula_9 it is called degenerate.
In what follows, unless otherwise stated we'll assume all Orlicz functions are nondegenerate. This implies formula_10 for all formula_11.
For each scalar sequence formula_12 set
We then define the Orlicz sequence space with respect to formula_14, denoted formula_15, as the linear space of all formula_12 such that formula_17 for some formula_18, endowed with the norm formula_19.
Two other definitions will be important in the ensuing discussion. An Orlicz function formula_14 is said to satisfy the Δ2 condition at zero whenever
We denote by formula_22 the subspace of scalar sequences formula_23 such that formula_17 for all formula_18.
Properties.
The space formula_15 is a Banach space, and it generalizes the classical formula_1 spaces in the following precise sense: when formula_28, formula_29, then formula_19 coincides with the formula_1-norm, and hence formula_32; if formula_14 is the degenerate Orlicz function then formula_19 coincides with the formula_35-norm, and hence formula_36 in this special case, and formula_37 when formula_14 is degenerate.
In general, the unit vectors may not form a basis for formula_15, and hence the following result is of considerable importance.
Theorem 1. If formula_14 is an Orlicz function then the following conditions are equivalent:
Two Orlicz functions formula_14 and formula_58 satisfying the Δ2 condition at zero are called equivalent whenever there exist are positive constants formula_59 such that formula_60 for all formula_9. This is the case if and only if the unit vector bases of formula_15 and formula_63 are equivalent.
formula_15 can be isomorphic to formula_63 without their unit vector bases being equivalent. (See the example below of an Orlicz sequence space with two nonequivalent symmetric bases.)
Theorem 2. Let formula_14 be an Orlicz function. Then formula_15 is reflexive if and only if
Theorem 3 (K. J. Lindberg). Let formula_70 be an infinite-dimensional closed subspace of a separable Orlicz sequence space formula_15. Then formula_70 has a subspace formula_73 isomorphic to some Orlicz sequence space formula_63 for some Orlicz function formula_58 satisfying the Δ2 condition at zero. If furthermore formula_70 has an unconditional basis then formula_73 may be chosen to be complemented in formula_70, and if formula_70 has a symmetric basis then formula_70 itself is isomorphic to formula_63.
Theorem 4 (Lindenstrauss/Tzafriri). Every separable Orlicz sequence space formula_15 contains a subspace isomorphic to formula_1 for some formula_29.
Corollary. Every infinite-dimensional closed subspace of a separable Orlicz sequence space contains a further subspace isomorphic to formula_1 for some formula_29.
Note that in the above Theorem 4, the copy of formula_1 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 formula_15 which fails to contain a complemented copy of formula_1 for any formula_90. This same space formula_15 contains at least two nonequivalent symmetric bases.
Theorem 5 (K. J. Lindberg & Lindenstrauss/Tzafriri). If formula_15 is an Orlicz sequence space satisfying formula_93 (i.e., the two-sided limit exists) then the following are all true.
Example. For each formula_29, the Orlicz function formula_100 satisfies the conditions of Theorem 5 above, but is not equivalent to formula_101.