Jolanta Grala-Michalak, Artur Michalak∗
On constructions of isometric copies of L
p(0, 1) spaces (0 < p 6 2) by stochastic p-stable processes
Abstract. Let Sp={Spt : t =2kn, 06 k 6 2n, n∈ N} be a stochastic process on a probability space (Ω, Σ, P ) with independent and time homogeneous increments such that Stp− Supis identically distributed as (t− u)1pZpfor each 06 u < t 6 1 where Zpis a given symmetric p-stable distribution. We show that the closed linear hull of Spforms an isometric copy of the real Lebesgue space Lp(0, 1) in any quasi-Banach space X consisting of P -a.e. equivalence classes of Σ-measurable real functions on Ω equipped with a rearrangement invariant quasi-norm which contains Spas a subset.
It is possible to construct processes Sp for 0 < p6 2 on [0, 1] with the Lebesgue measure. We show also a complex version of the result.
2000 Mathematics Subject Classification: Primary 46E30; Secondary 60J30, 46B09, 46B25.
Key words and phrases: Lp-spaces.
Johnson, Maurey, Schechtman and Tzafriri showed in [5] that if a symmetric Banach function space on [0, 1] contains a function t → t−1p for 1 < p < 2, then it contains an isometric copy of the Lebesgue space Lp(0, 1). The aim of this paper is to show a simple proof of the above fact based on properties of some stochastic processes. Our considerations are close to the classical results of Kadec, Kwapie´n and Kanter. In [6] Kadec showed that any sequence of independent symmetric p- stable distributions on a probability space (Ω, Σ, P ) spans an isometric copy Xpof the sequence space lp in the real Lebesgue space Lr(P ) for every 1 6 r < p < 2 (his considerations remain valid for every 0 < r < p < 2 (see [15, III.A.16]). For p = 2 the subspace Xp is isometric to l2 in Lq(P ) for every 0 < q < ∞. Let Sp = {Stp: 06 t 6 1} be a stochastic process on a probability space (Ω, Σ, P) with independent and time homogeneous p-stable increments. In [8, p. 197] Kwapie´n
∗The authors wish to thank Professor M. W´ojtowicz whose remarks allowed to improve the paper.
noted as a well known fact that for every 0 6 r < p < 2 the stochastic integral Tp(f) = R1
0 f (t) dStp defines an isomorphic embedding of Lp(0, 1) into Lr(P ). If p = 2 the same is true for each 06 r < ∞. A proof for the case 0 < r < p 6 2 is given in [7].
In this paper we show that the process Sp spans an isometric copy of real Lp(0, 1) in any quasi-Banach space X consisting of P -a.e. equivalence classes of Σ-measurable scalar functions equipped with a rearrangement invariant quasi-norm such that X contains Sp as a subset.
1. Preliminaries. The present paper deals with both real and complex quasi- Banach spaces, and the operators acting between such spaces are assumed to be linear. A quasi-Banach space X is a locally bounded complete vector space. Its topology is generated by a quasi-norm (the gauge functional of an open, symmetric, bounded set), i.e. a functional k · kX: X → [0, ∞) such that
kzkX = 0 if and only if z = 0, kaxkX = |a|kxkX,kx + ykX6 CX(kxkX+ kykX) for every scalar a, for all x, y ∈ X and some constant CX > 0. For more information on quasi-Banach spaces the reader is referred to [10]. The closed linear hull of a subset A in X is denoted by linA. We also apply the convention that every norm is a quasi-norm.
Throughout this paper the triple (Ω, Σ, P ) denotes a probability space and r ∈ (0, ∞). The Lebesgue space Lr(P ) consists of all (P -a.e. equivalence classes) of Σ- measurable scalar functions f on Ω such that kfkLr(P )= R
Ω|f|rdP1r
<∞. The space Lr(P ) equipped with the quasi-norm k·kLp(P )is quasi-Banach for 0 < r < ∞ and Banach for 1 6 r < ∞. The symbol λ stands for the Lebesgue measure on Ω = [0, 1], and then the Lebesgue space Lr(λ) is denoted by Lr(0, 1). A quasi- Banach space X consisting of (P -a.e. equivalence classes of) Σ-measurable scalar functions is called symmetric if for any Σ-measurable functions x and y the following conditions hold:
(i) if x ∈ X and |y| 6 |x| P -a.e., then y ∈ X and kykX6 kxkX,
(ii) if x ∈ X and P ({|x| > t}) = P ({|y| > t}) for every t > 0, then y ∈ X and kykX= kxkX.
We say that a quasi-norm k · kX on a quasi-Banach space X consisting of (P -a.e.
equivalence classes of) Σ-measurable scalar functions X is rearrangement invariant if kfkX = kgkX for every f, g ∈ X such that P ({|f| > t}) = P ({|g| > t}) for every t> 0. For a Σ-measurable function f : Ω → C the decreasing rearrangement f∗ : [0, 1] → R is defined as f∗(s) = inf{t > 0 : P ({|f| > t}) 6 s}. For more information on symmetric Banach spaces the reader is referred to [1], [5] and [9].
Proposition 1.1 If 0 < p <∞ and (en) is a sequence in a quasi-Banach space X such that
(1) en= e2n+ e2n+1 for every n,
(2) there exist constants 0 < c6 C such that for every n and scalars a2n,..., a2n+1−1
c2−np2n+1X−1
k=2n
|ak|p1p
6
2n+1X−1 k=2n
akek
X6 C2−np2n+1X−1
k=2n
|ak|p1p
,
then the subspace lin{en: n ∈ N} of X is isomorphic to the Lebesgue space Lp(0, 1).
Moreover, if c = C the subspace is isometric to Lp(0, 1).
For p = 1, a slightly modified version of the above proposition is addressed in [2, p. 211, Exercise 2].
Proof Let A1= (0, 1] and A2n+k= (2kn,k+12n ] for every 06 k 6 2n− 1 and n ∈ N.
The sets A2n, . . . , A2n+1−1 are pairwise disjoint and A2n∪ A2n+1 = An for every n ∈ N. It is clear that χAn are elements of Lp(0, 1). Moreover, for every n and scalars a2n, . . . , a2n+1−1
2n+1X−1 k=2n
akχAk
Lp(0,1)= 1 2n
2n+1X−1 k=2n
|ak|p1p .
Let Fn be the linear hull of the set {χA2n, . . . , χA2n+1−1} in Lp(0, 1). It is clear that the operator Ln: Fn→ X given by
Ln
2n+1X−1
k=2n
akχAk
=
2n+1X−1 k=2n
akek
is well defined and ckfkLp(0,1)6 kLn(f)kX 6 CkfkLp(0,1)for every f ∈ Fn. More- over, Lm|Fn = Ln for every m > n. Let an operator L :S∞
n=1Fn → X be given by the formula L|Fn = Tn. It is clear that L is well defined and ckfkLp(0,1) 6 kL(f)kX 6 CkfkLp(0,1)for every f ∈S∞
n=1Fn. Since X is complete and S∞n=1Fn
is a dense subset of Lp(0, 1), there exists a continuous extension ˜L of L which is the
isomorphism we are looking for.
We will also apply some probabilistic notations. Let (Ω1, Σ1, P1) be also a probability space. We say that a Rn-valued Σ-measurable function f on Ω has the same distribution as a Rn-valued Σ1-measurable function g on Ω1 if P (f−1(B)) = P1(g−1(B)) for every Borel subset B of Rn. It is clear that if two Σ-measurable functions f, g : Ω → C has the same distribution, then P ({|f| > t}) = P ({|g| > t}) for every t > 0. Moreover for any Σ-measurable function f : Ω → C the function f∗ has the same distribution as |f|. For a random variable f = (f1, . . . , fn) on (Ω, Σ, P ) taking values in Rn the characteristic function ϕf : Rn → C is defined as
ϕf((t1, . . . , tn)) =Z
Ω
eiPnj=1tjfjdP
for every (t1, . . . , tn) ∈ Rn. We say that Σ-measurable functions f1, . . . , fn: Ω → Rk are independent if for every Borel subsets A1, . . . , An of Rk the following equality
holds P (f1−1(A1) ∩ · · · ∩ fn−1(An)) = P (f1−1(A1)) · . . . · P (fn−1(An)). For more information on probabilistic notions applied in this paper the reader is referred to [3], [4] and [14].
For every 0 < p6 2 let Zpbe a random variable on (Ω, Σ, P ) with the character- istic function of the form ϕZp(t) = exp(−|t|p) for every t ∈ R. It is well known that such random variables and probability spaces do exist (see [15, Theorem III.A.14]).
Let νpdenotes the probability Borel measure on R of the form νp(A) = P (Zp−1(A))
for every Borel subset A of R. The measure νpis absolutely continuous with respect to the Lebesgue measure on R (the reader may find the density function of νp in [4, XVII.6]). In the sequel we will need the following fact (see [15, III.A.16]).
Proposition 1.2 Let p and Zp be defined as above. If f1, . . . , fn are indepen- dent random variables on a probability space with the same distribution as Zp and a1, . . . , an ∈ R, then the random variable Pn
k=1akfk has the same distribution as
Pn
k=1|ak|pp1 Zp.
In the complex case we need for every 0 < p 6 2 a pair Vp = (Vp1, Vp2) of random variables, i.e. Vp is a random variable taking values in R2, on (Ω, Σ, P ) with the characteristic function of the form ϕVp(t1, t2) = exp −(t21+ t22)p2for every (t1, t2) ∈ R2. It is well known that such random variables and probability spaces do exist (see [14, p. 193-200]). Let µp denotes the probability Borel measure on R2 of the form
µp(A) = P (Vp−1(A)) (#)
for every Borel subset A of R2. The random variables Vp1 and Vp2 have the same distributions as Zp (of course Vp1 and Vp2are not independent).
In the sequel we will apply the following fact.
Proposition 1.3 Let p and Vp be defined as above. If f1, . . . , fn are independent random variables taking values in R2on a probability space with the same distribu- tion as Vp and a1, . . . , an ∈ C, then the random variablePn
k=1akfk has the same distribution as Pn
k=1|ak|pp1
Vp.
This result seems to be known, for the sake of completeness we present its proof.
Proof First note that for every linear isometry I : R2 → R2 we have equality ϕVp= ϕVp◦ I. Let bω be the Fourier transform of a Borel measure ω onR2. If ω is a probability Borel measure on R2and L : R2→ R2is a linear map, then
c
ωL(u) = bω(L∗(u))
for every u ∈ R2, where ωL is the probability Borel measure on R2 given by the formula ωL(B) = ω(L−1(B)) for every Borel subset B of R2 and L∗ is the adjoint
operator of L. Combining these observations with the the fact that the Fourier transform on the space of Borel measures on R2is an injective map for probability measure µp defined in (#) we obtain
µp(I−1(B)) = µp(B)
for every Borel subset B of R2 and for every isometry I : R2→ R2. It shows that for every b ∈ C with |b| = 1 the random variable bVp has the same distribution as Vp. Hence ϕbVp = ϕVp. Therefore
ϕaVp(t) = ϕ|a|Vp(t) = ϕVp(|a|t)
for every a, t ∈ C. Let a1, . . . , an ∈ C. Since the random variables a1f1, . . . , anfn
are independent, the characteristic function of Pnk=1akfk has the form
ϕPnk=1akfk(t1, t2) = exp
−Xn
k=1
|ak|p
(t21+ t22)p2
= ϕ(Pn
k=1|ak|p)1pVp
(t1, t2)
for every (t1, t2) ∈ R2. Applying once again the fact that the Fourier transform on the space of Borel measures on R2 is an injective map, we obtain that the random variable Pnk=1akfk has the same distribution asPn
k=1|ak|p1p
Vp, as claimed. A stochastic process on (Ω, Σ, P ) is any family {ft : t ∈ T } of Rn-valued Σ- measurable functions on Ω, where T is a given subset of R. Let, for every t ∈ T , the symbol πt denote the projection (Rn)T → Rn defined by the formula πt(x) = x(t).
The σ-algebra generated by {πt : t ∈ T } is denoted by B(Rn)T. Every stochastic process {ft : t ∈ T } defines a probability measure on ((Rn)T,B(Rn)T) such that the processes {ft : t ∈ T } and {πt : t ∈ T } have the same finite dimensional distributions. A stochastic process {ft : t ∈ T } has independent increments if for every t1, . . . , tn ∈ T such that t1 < · · · < tn the random variables ft1, ft2 − ft1, . . . , ftn− ftn−1 are independent.
The convolution of two Borel measures η and ω on Rn is a Borel measure on Rn defined as η ∗ ω(A) = R
Rnη(A− x) dω(x) for every Borel subset A of Rn. A probability measure on Σ is said to be atomless if every set E ∈ Σ such that ν(E) > 0 contains a subset A∈ Σ with 0 < ν(A) < ν(E).
Proposition 1.4 Let p, Zp, and Vp have the same meaning as in Propositions 1.2 and 1.3, and put Q := {2kn : 1 6 k 6 2n, n∈ N} and Q0:= {2kn : 06 k 6 2n, n∈ N}.a) There exists an atomless probability Borel measure ωp on RQ such that the process Sp = {πt : t ∈ Q0}, where π0 = 0, has the following properties: (1) ωp(π0= 0) = 1, (2) Sphas independent increments, (3) the random variable πt−πu has the same distribution as (t − u)1pZp for every 06 u < t 6 1.
b) There exists an atomless probability Borel measure ωp on (R2)Q such that the process Sp = {πt : t ∈ Q0}, where π0 = 0, has the following properties: (1) ωp(π0 = 0) = 1, (2) Sp has independent increments, (3) the random variables πt− πu has the same distribution as (t − u)1pVp for every 06 u < t 6 1.
Proof For a > 0, let νp,a denote the probability Borel measure on R of the form νp,a(A) = P ((aZp)−1(A))
for every Borel subset A of R. From Proposition 1.2 it follows that the family N = {νp,a1p : a > 0} forms a semigroup of probability Borel measures on R, i.e.,
νp,a1p ∗ νp,bp1 = ν
p,(a+b)1p
for every a, b > 0. The semigroup generates a probability measure ηp on the space (R(0,∞),BR(0,∞)) such that the projections {πt : t > 0}, where π0 = 0, form a stochastic process on (R(0,∞),BR(0,∞), ηp) with independent increments, and
(πt− πu)−1(A) = νp,t−u(A) (&)
for every Borel subset A of R and for every pair t, u ∈ [0, ∞) with u < t (this follows from [13, Theorem 7.14]).
Let ξQ denote the surjection from R(0,∞)onto RQ of the form ξQ(x) = x|Q.
Since ξQ is (R(0,∞),BR(0,∞)) - (RQ,BRQ) measurable, the formula ωp(A) = ηp(ξQ−1(A)) for every A ∈ BRQ
defines a probability Borel measure on RQ. Now set Sp = {πt : t ∈ Q0}, where π0= 0. It is a stochastic process on (RQ,BRQ, ωp), which fulfills condition (1) (as π0≡ 0) and by the properties of N (see (&)) the process fulfills also conditions (2) and (3).
b) The above considerations remain also valid for the semigroup M = {µ
p,a1p : a > 0} of probability Borel measures on R2, where the measure µp,a is defined by the formula
µp,a(A) = P ((aVp)−1(A))
for every Borel subset A of R2.
We will need the following consequence of Proposition 1.4.
Corollary 1.5 Let p, Zp, Vp, Q and Q0 have the same meaning as in Proposi- tion 1.4, and let (Ω, Σ, P ) be an atomless probability space.
a) There exists a real-valued stochastic process Sp= {Stp: t ∈ Q0} on (Ω, Σ, P ) with the following properties: (1) P ({S0p= 0}) = 1, (2) Sp has independent incre- ments, (3) the random variable Stp− Sup has the same distribution as (t − u)1pZp for every 06 u < t 6 1.
b) There exists an R2-valued stochastic process Sp = {Stp: t ∈ Q0} on (Ω, Σ, P ) with the following properties: (1) P ({S0p = 0}) = 1, (2) Sp has independent in- crements, (3) the pair of random variables Stp− Sup has the same distribution as (t − u)1pVp for every 06 u < t 6 1.
Proof We shall prove only case a), because our arguments remain valid also in case b).
Let ωp be the atomless probability Borel measure on RQ taken from Proposi- tion 1.4 a). The space RQ is a complete separable metric space and BRQ coincides with the σ-algebra of Borel subsets of RQ. According to [11, Theorem 15.3.9] there exist Borel subsets A of [0, 1] and B of RQ with λ(A) = ωp(B) = 1 and a Borel equivalence f0: A → B (i.e., f0takes Borel sets into Borel sets and its inverse takes Borel sets into Borel sets) such that
λ(C) = ωp(f0(C ∩ A))
for every Borel subset C of [0, 1]. It is easy to check that we also have λ(f0−1(C ∩ B)) = ωp(C) for every Borel subset C of RQ. Now we shall define three auxiliary functions f, g, h such that the process {πt◦ f ◦ g ◦ h : t ∈ Q0} possesses the required properties (1), (2) and (3).
Let f : [0, 1] → RQ be the map given by the formula
f (x) =
(f0(x) if x ∈ A 0 if x ∈ [0, 1] \ A.
Let ({0, 1}N,B{0,1}N, η) be the Cantor set with the σ-algebra of Borel subsets and the Haar measure (i.e., the product space ({0, 1}, 2{0,1},12δ0+12δ1)N where δ0
and δ1 are Dirac measures concentrated at 0 and 1, respectively). It is well known that the map g : {0, 1}N→ [0, 1] given by the formula
g((n)) =X∞
n=1
n
2n,
for every (εn) ∈ {0, 1}N, is Borel measurable and η(g−1(C)) = λ(C) for every Borel subset C of [0, 1].
By the Lapunov theorem (see [12, Theorem 5.5]), there is a sequence (Dn) in Σ such that D1 = Ω, P (D2n+k) = 2−n for every 0 6 k < 2n and n ∈ N, D2n+k∩ D2n+l = ∅ for every 0 6 k < l < 2n, D2n∪ D2n+1 = Dn for every n ∈ N.
Set
ψn(x) =
(1 if x ∈S2n−1
j=1 D2n+2j−1
0 if x /∈S2n−1
j=1 D2n+2j−1. It is easy to see that P (E ∩S2n−1
j=1 D2n+2j−1) = 12P (E) for every member E of the σ-algebra generated by {Dj : 1 6 j < 2n}. Therefore (ψn) is a sequence of independent random variables on (Ω, Σ, P ) with the Bernoulli distribution B(1,12).
Hence the map h : Ω → {0, 1}N given by the formula h(x) = (ψn(x))
is Σ - B{0,1}N measurable and P (h−1(C)) = η(C) for every Borel subset C of {0, 1}N.
For t ∈ Q0, we put Stp := πt◦ f ◦ g ◦ h. Now we shall check that the process {Stp: t ∈ Q0} has properties (1), (2) and (3). First let us note that for every Borel subset B of R we have
P ((f◦ g ◦ h)−1(B)) = η(g−1(f−1(B))) = λ(f−1(B)) = ωp(B).
The property (1) is a straightforward consequence of the equality P ((π0◦ f ◦ g ◦ h)−1(B)) = ωp(π0−1(B)), that holds for every Borel subset B of R.
For the proof of property (2) notice first that for every 06 t1<· · · < tn6 1 the random variables πt1, πt2−πt1, . . . , πtn−πtn−1on the probability space (RQ,BRQ, ωp) are independent. Hence for every Borel subsets B1, . . . Bn of R we obtain the fol- lowing equalities:
P ((πt1◦ f ◦ g ◦ h)−1(B1) ∩
\n k=2
((πtk− πtk−1) ◦ f ◦ g ◦ h)−1(Bk))
= ωp(πt−11 (B1) ∩
\n k=2
(πtk− πtk−1)−1(Bk))
= ωp(πt−11 (B1)) · Yn k=2
ωp((πtk− πtk−1)−1(Bk))
= P ((πt1◦ f ◦ g ◦ h)−1(B1)) · Yn k=2
P (((πtk− πtk−1) ◦ f ◦ g ◦ h)−1(Bk)).
This shows that the process Sp has independent increments, as claimed.
Since for every 06 u < t 6 1 and for every Borel subset B of R we have equality P (((πt− πu) ◦ f ◦ g ◦ h)−1(B)) = ωp((πt− πu)−1(B)),
the random variable πt◦ f ◦ g ◦ h − πu◦ f ◦ g ◦ h on the probability space (Ω, Σ, P ) has the same distribution as the random variable πt− πu on the probability space
(RQ,BRQ, ωp). This proves (3).
2. Main results. In this section, the symbols p, Q0, Zp, Vp have the same meaning as in Proposition 1.5.
Theorem 2.1 Let 0 < p6 2. Let (Ω, Σ, P) be an atomless probability space. Let X be a quasi-Banach space consisting of (P -a.e. equivalence classes of) Σ-measurable scalar functions with rearrangement invariant quasi-norm and let Sp = {Stp : t ∈ Q0} be a stochastic process with properties (1), (2), (3) of Corollary 1.5, a) or b), respectively, for X real or complex, respectively.
If X contains Sp as a subset and kS1pkX > 0, then the subspace lin{Spt : t ∈ Q0} of X is isometric to Lp(0, 1).
Proof By our assumptions, the elements e1= S1p= S1p− S0p and e2n+k= Spk+1 2n − Spk
2n are members of X for every 06 k < 2nand n ∈ N. It is clear that (en) satisfies property (1) of Proposition 1.1.
First we consider the real case. By part (2) of Corollary 1.5 a), the elements e2n = Sp1
2n, e2n+1 = Sp2
2n − Sp2n1 , . . . , e2n+1−1 = S1p− S1p−2n1 are independent ran- dom variables with the same distribution as 2−npZp. According to Proposition 1.2 for every a1, . . . , an ∈ R the function P2n+1−1
k=2n akek has the same distribution as Pn
k=1|2−npak|p1pS1p. Since the quasi-norm on X is rearrangement invariant, we
have
2n+1X−1 k=2n
akek
X= 2−np2n+1X−1
k=2n
|ak|p1p ke1kX.
Consequently, (en) satisfies the property (2) of Proposition 1.1 with c = C = kS1pkX. An appeal to Proposition 1.1 completes the proof for the real case.
Now let us consider the complex case. By part (2) of Corollary 1.5 b), the elements e2n= Sp1
2n, e2n+1= Sp2
2n−Sp2n1 , . . . , e2n+1−1 = S1p−S1−p 2n1 are independent random variables taking values in R2 with the same distribution as 2−npVp. By Proposition 1.3, for every a1, . . . , an ∈ C the functionP2n+1−1
k=2n akek has the same distribution as Pnk=1|2−npak|pp1
S1p. Since the quasi-norm on X is rearrangement invariant, we have
2n+1X−1 k=2n
akek
X= 2−np2n+1X−1
k=2n
|ak|p1p ke1kX.
Consequently, (en) satisfies the property (2) of Proposition 1.1 with c = C = kS1pkX. An appeal to Proposition 1.1 completes the proof for the complex space. According to [4, Exercise VIII.29] for every 0 < p < 2 there exists a constant Cp > 0 such that νp(|s| > t) 6 Cpt−p for every t > 0. Hence the decreasing rearrangement of S1p has the following property: (S1p)∗(t)6 Cp1pt−1p for every t > 0.
For p = 2 we have
ν2(|s| > t) = 2√1πZ
|s|>texp(−s42)ds6 exp(−t42) for every t > 0. It follows that (Sp1)∗(t) 6 2p
| ln(t)| for every t > 0. Similar inequalities are also valid for the measure µp. As a straightforward consequence of Theorem 2.1 and the above estimations we obtain the following corollary.
Corollary 2.2 If a symmetric quasi-Banach function space on [0, 1] contains the function t → t−1p for some 0 < p < 2, then it contains an isometric copy of the Lebesgue space Lp(0, 1).
If a symmetric quasi-Banach function space on [0, 1] contains the function t → p| ln(t)|, then it contains an isometric copy of the Lebesgue space L2(0, 1).
References
[1] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, Inc., Boston 1988.
[2] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, 92 Springer-Verlag, New York 1984.
[3] G. A. Edgar, Integral, Probability and Fractal Measures, Springer-Verlag, New York 1998.
[4] W. Feller, An Introduction to Probability Theory and its Applications Vol. II, John Wiley &
Sons, Inc., New York 1971.
[5] W. B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structure in Banach spaces, Mem. Amer. Math. Soc. 217, 1979.
[6] M. I. Kadec, On linear dimension of the spaces Lpand lq, Uspiekhi Mat. Nauk. 13(6) (1958), 95-98 (in Russian).
[7] M. Kanter, Stable laws and the imbedding of Lp spaces, Amer. Math. Monthly 80 (1973), 403-407.
[8] S. Kwapie´n, On a theorem of L. Schwartz and its applications to absolutely summing opera- tors, Studia Math. 38 (1970), 193-201.
[9] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II, Ergebnisse der Mathematik und ihrer Grenzgebite 97, Springer-Verlag, Berlin 1979.
[10] S. Rolewicz, Metric Linear Spaces, PWN Warszawa 1984.
[11] H. L. Royden, Real Analysis, The Macmillan Company, New York 1968.
[12] W. Rudin, Functional Analysis, McGraw-Hill, Inc., New York 1991.
[13] K. R. Stromberg, Probability for analysts, Chapman & Hall, New York 1994.
[14] V. Uchaikin, V. Zolotarev, Chance and Stability, Modern Probability and Statistics, VSP, Utrecht 1999.
[15] P. Wojtaszczyk, Banach spaces for analysts, Cambridge University Press, Cambridge 1991.
Jolanta Grala-Michalak
Faculty of Mathematics and Computer Science, Adam Mickiewicz University ul. Umultowska 87, 61-614 Pozna´n, Poland
E-mail: grala@amu.edu.pl Artur Michalak
Faculty of Mathematics and Computer Science, Adam Mickiewicz University ul. Umultowska 87, 61-614 Pozna´n, Poland
E-mail: michalak@amu.edu.pl
(Received: 9.01.2007)
