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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 ℓ p {\displaystyle \ell _{p}} spaces, and as such play an important role in functional analysis. Orlicz sequence spaces are particular examples of Orlicz spaces.

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Definition

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

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

‖ ( a n ) n = 1 ∞ ‖ M = inf { ρ > 0 : ∑ n = 1 ∞ M ( | a n | / ρ ) ⩽ 1 } . {\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 M {\displaystyle M} , denoted ℓ M {\displaystyle \ell _{M}} , as the linear space of all ( a n ) n = 1 ∞ ∈ K N {\displaystyle (a_{n})_{n=1}^{\infty }\in \mathbb {K} ^{\mathbb {N} }} such that ∑ n = 1 ∞ M ( | a n | / ρ ) < ∞ {\textstyle \sum _{n=1}^{\infty }M(|a_{n}|/\rho )<\infty } for some ρ > 0 {\displaystyle \rho >0} , endowed with the norm ‖ ⋅ ‖ M {\displaystyle \|\cdot \|_{M}} .

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

lim sup t → 0 M ( 2 t ) M ( t ) < ∞ . {\displaystyle \limsup _{t\to 0}{\frac {M(2t)}{M(t)}}<\infty .}

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

Properties

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

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

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

  1. M {\displaystyle M} satisfies the Δ2 condition at zero, i.e. lim sup t → 0 M ( 2 t ) / M ( t ) < ∞ {\textstyle \limsup _{t\to 0}M(2t)/M(t)<\infty } .
  2. For every λ > 0 {\displaystyle \lambda >0} there exists positive constants K = K ( λ ) {\displaystyle K=K(\lambda )} and b = b ( λ ) {\displaystyle b=b(\lambda )} so that M ( λ t ) ⩽ K M ( t ) {\displaystyle M(\lambda t)\leqslant KM(t)} for all t ∈ [ 0 , b ] {\displaystyle t\in [0,b]} .
  3. lim sup t → 0 t M ′ ( t ) / M ( t ) < ∞ {\textstyle \limsup _{t\to 0}tM'(t)/M(t)<\infty } (where M ′ {\displaystyle M'} 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. ℓ M = h M {\displaystyle \ell _{M}=h_{M}} .
  5. The unit vectors form a boundedly complete symmetric basis for ℓ M {\displaystyle \ell _{M}} .
  6. ℓ M {\displaystyle \ell _{M}} is separable.
  7. ℓ M {\displaystyle \ell _{M}} fails to contain any subspace isomorphic to ℓ ∞ {\displaystyle \ell _{\infty }} .
  8. ( a n ) n = 1 ∞ ∈ ℓ M {\displaystyle (a_{n})_{n=1}^{\infty }\in \ell _{M}} if and only if ∑ n = 1 ∞ M ( | a n | ) < ∞ {\textstyle \sum _{n=1}^{\infty }M(|a_{n}|)<\infty } .

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

ℓ M {\displaystyle \ell _{M}} can be isomorphic to ℓ N {\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 M {\displaystyle M} be an Orlicz function. Then ℓ M {\displaystyle \ell _{M}} is reflexive if and only if

lim inf t → 0 t M ′ ( t ) M ( t ) > 1 {\displaystyle \liminf _{t\to 0}{\frac {tM'(t)}{M(t)}}>1\;\;} and lim sup t → 0 t M ′ ( t ) M ( t ) < ∞ {\displaystyle \;\;\limsup _{t\to 0}{\frac {tM'(t)}{M(t)}}<\infty } .

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

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

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

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

Theorem 5 (K. J. Lindberg & Lindenstrauss/Tzafriri). If ℓ M {\displaystyle \ell _{M}} is an Orlicz sequence space satisfying lim inf t → 0 t M ′ ( t ) / M ( t ) = lim sup t → 0 t M ′ ( t ) / M ( t ) {\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.

  1. ℓ M {\displaystyle \ell _{M}} is separable.
  2. ℓ M {\displaystyle \ell _{M}} contains a complemented copy of ℓ p {\displaystyle \ell _{p}} for some 1 ⩽ p < ∞ {\displaystyle 1\leqslant p<\infty } .
  3. ℓ M {\displaystyle \ell _{M}} has a unique symmetric basis (up to equivalence).

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