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 spaces, and as such play an important role in functional analysis.
Fix so that denotes either the real or complex scalar field. We say that a function is an Orlicz function if it is continuous, nondecreasing, and (perhaps nonstrictly) convex, with and . In the special case where there exists with for all it is called degenerate.
In what follows, unless otherwise stated we'll assume all Orlicz functions are nondegenerate. Note that this implies for all .
For each scalar sequence set
We then define the Orlicz sequence space with respect to , denoted , as the linear space of all such that for some , endowed with the norm .
Two other definitions will be important in the ensuing discussion. An Orlicz function is said to satisfy the Δ2 condition at zero whenever
We denote by the subspace of scalar sequences such that for all .
The space is a Banach space, and it generalizes the classical spaces in the following precise sense: when , , then coincides with the -norm, and hence ; if is the degenerate Orlicz function then coincides with the -norm, and hence in this special case. Note also that when is degenerate.
In general, the unit vectors may not form a basis for , and hence the following result is of considerable importance.
Theorem 1. If is an Orlicz function then the following conditions are equivalent:
- (i) satisfies the Δ2 condition at zero, i.e. .
- (ii) For every there exists positive constants and so that for all .
- (iii) (where 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).
- (iv) .
- (v) The unit vectors form a boundedly complete symmetric basis for .
- (vi) is separable.
- (vii) fails to contain any subspace isomorphic to .
- (viii) if and only if .
Two Orlicz functions and satisfying the Δ2 condition at zero are called equivalent whenever there exist are positive constants such that for all . This is the case if and only if the unit vector bases of and are equivalent.
Note that can be isomorphic to without their unit vector bases being equivalent. (See the example below of an Orlicz sequence space with two nonequivalent symmetric bases.)
Theorem 2. Let be an Orlicz function. Then is reflexive if and only if
- and .
Theorem 3 (K. J. Lindberg). Let be an infinite-dimensional closed subspace of a separable Orlicz sequence space . Then has a subspace isomorphic to some Orlicz sequence space for some Orlicz function satisfying the Δ2 condition at zero. If furthermore has an unconditional basis then may be chosen to be complemented in , and if has a symmetric basis then itself is isomorphic to .
Theorem 4 (Lindenstrauss/Tzafriri). Every separable Orlicz sequence space contains a subspace isomorphic to for some .
Corollary. Every infinite-dimensional closed subspace of a separable Orlicz sequence space contains a further subspace isomorphic to for some .
Note that in the above Theorem 4, the copy of 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 which fails to contain a complemented copy of for any . This same space contains at least two nonequivalent symmetric bases.
Theorem 5 (K. J. Lindberg & Lindenstrauss/Tzafriri). If is an Orlicz sequence space satisfying (i.e., the two-sided limit exists) then the following are all true.
- (i) is separable.
- (ii) contains a complemented copy of for some .
- (iii) has a unique symmetric basis (up to equivalence).
Example. For each , the Orlicz function satisfies the conditions of Theorem 5 above, but is not equivalent to .
- Lindenstrauss, J., and L. Tzafriri. Classical Banach Spaces I, Sequence Spaces (1977), ISBN 978-3-642-66559-2.
- Lindenstrass, J., and L. Tzafriri. "On Orlicz Sequence Spaces," Israel Journal of Mathematics 10:3 (Sep 1971), pp379-390.
- Lindenstrass, J., and L. Tzafriri. "On Orlicz sequence spaces. II," Israel Journal of Mathematics 11:4 (Dec 1972), pp355-379.
- Lindenstrass, J., and L. Tzafriri. "On Orlicz sequence spaces III," Israel Journal of Mathematics 14:4 (Dec 1973), pp368-389.