Georgian Mathematical Journal Volume 9 (2002), Number 2, 295–301

GEOMETRY OF MODULUS SPACES

R. KHALIL, D. HUSSEIN, AND W. AMIN

Abstract. Let ϕ be a modulus function, i.e., continuous strictly increasing function on [0, ∞), such that ϕ(0) = 0, ϕ(1) = 1, and ϕ(x + y) ≤ ϕ(x) + ϕ(y) for all x, y in [0, ∞). It is the object of this paper to characterize, for any Banach space X, extreme points, exposed points, and smooth points of the unit ball of the metric linear space ℓ^{ϕ}(X), the space of all sequences (x_{n}), x_{n} ∈ X, n = 1, 2, …, for which ∑ ϕ(∥x_{n}∥) < ∞. Further, extreme, exposed, and smooth points of the unit ball of the space of bounded linear operators on ℓ^{p}, 0 < p < 1, are characterized.

2000 Mathematics Subject Classification: Primary: 47B38; Secondary: 48A65. Key words and phrases: Extreme point, exposed point, smooth point, modulus function.

0. Introduction. Let ϕ : [0, ∞) ⟶ [0, ∞) be a continuous function. We call ϕ a modulus function if:

(i) ϕ(x) = 0 if and only if x = 0; (ii) ϕ is increasing; (iii) ϕ(x + y) ≤ ϕ(x) + ϕ(y).

The functions ϕ(x) = x^{p}, p ∈ (0, 1), and ϕ(x) = ln(1 + x) are modulus functions.

For a modulus function ϕ, we let ℓ^{ϕ} denote the space of all real-valued sequences (x_{n}) for which ∑ ϕ(|x_{n}|) < ∞. For x, y ∈ ℓ^{ϕ}, d(x, y) = ∑ ϕ(|x_{n} − y_{n}|) is a metric on ℓ^{ϕ}. For x ∈ ℓ^{ϕ} we let ∥x∥_{ϕ} denote d(x, 0). The space (ℓ^{ϕ}, ∥∥_{ϕ}) is a metric linear space. These spaces were initiated by Ruckle [4].

Throughout this paper, R denotes the set of real numbers. If X is a Banach space, X^{∗} will denote the dual of X. If x^{∗} ∈ X^{∗} and x ∈ X, we let ⟨x^{∗}, x⟩ denote the value of x^{∗} at x. We let ℓ^{p} denote the space of all (real) sequences (x_{n}) for which ∑ |x_{n}|^{p} < ∞, 0 < p < ∞. For x ∈ ℓ^{p}, we let

∥x∥_{p} = \begin{cases}(∑ |x_{i}|^{p})^{\frac{1}{p}} & if 1 ≤ p < ∞, \\ ∑ |x_{i}|^{p} & if 0 < p < 1. \end{cases}

So ∥x∥_{1} is the 1-norm of x in ℓ^{1}. For p = ∞, ℓ^{∞} is the space of all bounded (real) sequences. If x ∈ ℓ^{∞}, we let ∥x∥_{∞} = sup_{i} |x_{i}|.

Let us summarize the basic properties of (ℓ^{ϕ}, ∥ ∥_{ϕ}) in

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Theorem A. Let ϕ be any modulus function. Then:

(1) (ℓ^{ϕ}, ∥ ∥_{ϕ}) is a complete metric linear space.

(2) If ∥x∥_{ϕ} ≤ ϕ(a), then ∥x∥_{1} ≤ a.

(3) ℓ^{ϕ} ⊆ ℓ^{1}, and the inclusion map I : ℓ^{ϕ} ⟶ ℓ^{1} is continuous.

(4) If ϕ(1) = 1, then for every x ∈ ℓ^{ϕ} there exists r > 0 such that ∥rx∥_{ϕ} = 1.

(5) There exist α and a in [0, ∞) such that ϕ(x) > αx for all x ∈ [0, a).

Proof. The proof of (1) is in [4]. Statements (2) and (3) are easy to handle. Statement (5) is in [5]. So we prove only (4).

There are two cases:either ∥x∥_{ϕ} < 1 or ∥x∥_{ϕ} > 1. If ∥x∥_{ϕ} > 1, define F : [0, 1] ⟶ [0, ∞) by F (t) = ∥tx∥_{ϕ}. Then F is continuous with F (0) = 0 and F (1) > 1. By the intermediate value theorem there is r ∈ (0, 1) such that F (r) = 1. Hence ∥rx∥_{ϕ} = 1. The other case follows from statement (2) and the assumption ϕ(1) = 1.

Let X be a Banach space. A linear mapping T:ℓ^{ϕ} ⟶ X is called bounded if there exists λ > 0 such that ∥T x∥ ≤ λ for all x in ℓ^{ϕ} for which ∥x∥_{ϕ} ≤ 1. We let L(ℓ^{ϕ}, X) denote the space of all bounded linear operators on ℓ^{ϕ} with values in X. We let (ℓ^{ϕ})^{∗} denote L(ℓ^{ϕ}, R). For T ∈ L(ℓ^{ϕ}, X) we set ∥T ∥ = sup{∥T x∥ : ∥x∥_{ϕ} ≤ 1}. For the case 0 < p < 1 we let B(ℓ^{p}, ℓ^{p}) denote the space of linear operators on ℓ^{p} for which ∥T x∥_{p} ≤ λ ∥x∥_{p} for all x ∈ ℓ^{p}

with some λ depending on T. Since a|b|^{p} = |a^{\frac{1}{p}} b|^{p} for a > 0, it follows that sup{∥T x∥_{p} : ∥x∥_{p} ≤ 1} = inf{λ : ∥T x∥_{p} ≤ λ ∥x∥_{p} for all x ∈ ℓ^{p}}. Hence B(ℓ^{p}, ℓ^{p}) = L(ℓ^{p}, ℓ^{p}).

For a modulus function ϕ and a Banach space X, we set ℓ^{ϕ}(X) = {(x_{n}) : x_{n} ∈ X and ∑ ϕ(∥x_{n}∥) < ∞}.If x = (x_{n}) ∈ ℓ^{ϕ}(X), then we define ∥x∥_{ϕ} = ∑ ϕ(∥x_{n}∥). It is easy to check that (ℓ^{ϕ}(X), ∥∥_{ϕ}) is a complete metric linear space.

Extreme points of the unit ball of L(ℓ^{p}, ℓ^{p}), 1 < p < ∞, have been studied extensively by many authors ([6]–[10] and others). A full characterization of extreme points of the unit ball of L(ℓ^{p}, ℓ^{p}), 1 < p < ∞, is still an open problem.

In this paper we characterize extreme,exposed, and smooth points of the unit balls of ℓ^{ϕ}, ℓ^{ϕ}(X) and L(ℓ^{p}, ℓ^{p}), 0 < p < 1.

1. Basic Structure of Spaces ℓ^{ϕ}(X). Throughout this paper we will assume that:

(i) ϕ is strictly increasing; (ii) ϕ(1) = 1.

Let M denote the class of all modulus functions satisfying (i) and (ii). We set (ℓ^{ϕ}(X))^{∗} = L(ℓ^{ϕ}, R), where R is the set of real numbers.

Theorem 1.1. Let ϕ ∈ M and X be any Banach space. Then [ℓ^{ϕ}(X)]^{∗} is isometrically isomorphic to ℓ^{∞}(X^{∗}).

Proof. Let F ∈ ℓ^{∞}(X^{∗}). So F = (x_{1}^{∗}, x_{2}^{∗}, …) with x_{i}^{∗} ∈ X^{∗} and sup_{i} ∥x_{i}^{∗}∥ < ∞.

Define F̃ : ℓ^{ϕ}(X) ⟶ R such that for x = (x_{i}) ∈ ℓ^{ϕ}(X), F̃(x) = ∑⟨x_{i}, x_{i}^{∗}⟩.

Hence |F̃(x)| ≤ ∑ ∥x_{i}∥ ∥x_{i}^{∗}∥ ≤ ∥F ∥ ∑ ∥x_{i}∥. Now for any function ϕ in M one can easily show that ℓ^{ϕ}(X) ⊆ ℓ^{1}(X). Further, if ∥f∥_{ϕ} = 1, then ∥f∥_{1} ≤ 1. Thus

∥F̃∥ ≤ ∥F ∥  (∗)

On the other hand, if F̃ ∈ [ℓ^{ϕ}(X)]^{∗}, then we define x_{i}^{∗} in X^{∗} as : x_{i}^{∗}(x) = F̃(0, 0, …, 0, x, 0, …) where x appears in the ith coordinate. Set F =(x_{1}^{∗}, x_{2}^{∗}, …). Then since sup_{i} ∥x_{i}^{∗}∥ ≤ ∥F̃∥, we obtain F ∈ ℓ^{∞}(X^{∗}) and ∥F ∥_{∞} ≤ ∥F̃∥ . This

together with (∗) gives ∥F ∥_{∞} = ∥F̃∥ . Thus the mapping J : ℓ^{∞}(X^{∗}) ⟶ [ℓ^{ϕ}(X)]^{∗}, J(F) = F̃ is linear onto and an isometry. This ends the proof.

As a consequence we get

Corollary 1.2. (ℓ^{ϕ})^{∗} = ℓ^{∞}.

Remark 1. If ϕ(x+y) < ϕ(x)+ϕ(y) for any x > 0, y > 0, then there are some elements x of ℓ^{ϕ} such that there is no x^{∗}in ℓ^{∞} for which ⟨x, x^{∗}⟩ = ∥x∥ ∥x^{∗}∥ . Indeed, if ∥x∥_{ϕ} = 1, then the continuity of ϕ, being strictly increasing and ϕ(1) = 1, implies that ∥x∥_{1} = 1 unless x has only one nonzero coordinate. So for x with more than one nonzero terms there cannot exist x^{∗} in ℓ^{∞} which attains its norm at x. However, if x has only one nonzero coordinate, then ∥x∥_{1} = ∥x∥_{ϕ} , if ∥x∥_{ϕ} = 1 and such x^{∗} exists.

2. Geometry of B_{1}(ℓ^{ϕ}(X)). A point x of a set K of a metric linear space E is called extreme if there exist no y and z in K such that y ≠ z and x = \frac{1}{2}(y + z). The point x in B_{1}(E) is called exposed if there exists f ∈ B_{1}(E^{∗}) such that f(x) = d(x, 0), and f(y) < d(y, 0) for all y in B_{1}(E), y ≠ x. We call x a smooth point of B_{1}(E) if there exists a unique f ∈ B_{1}(E^{∗}) such that f(x) = d(x, 0).

In this section we will characterize extreme, exposed, and smooth points of B_{1}(ℓ^{ϕ}(X)) for any Banach space X.

Theorem 2.1. Let ϕ ∈ M.The following statements are equivalent: (i) f is an extreme point of B_{1}(ℓ^{ϕ}(X)). (ii) f(n) = 0 for all n except for one coordinate, say, f(n_{0}), and f(n_{0})

is an extreme point of B_{1}(X).

Proof. (i)⟶(ii). Let f be extreme and, if possible, assume that f does not vanish at n_{1} and n_{2}. Define

g(n) = \begin{cases}f(n), & n ≠ n_{1}, n_{2}, \\ \frac{∥f(n_{1})∥+∥f(n_{2})∥}{∥f(n_{1})∥} f(n_{1}), & n = n_{1}, \\ 0, & n = n_{2}, \end{cases}

h(n) = \begin{cases}f(n), & n ≠ n_{1}, n_{2}, \\ \frac{∥f(n_{1})∥+∥f(n_{2})∥}{∥f(n_{2})∥} f(n_{2}), & n = n_{2}, \\ 0, & n = n_{1}. \end{cases}

Then g ≠ h. Further,

∥g∥_{ϕ} = ∑ ϕ(∥g(n)∥) ≤ ∑ ϕ ∥f(n)∥ ≤ 1.

Similarly, ∥h∥_{ϕ} ≤ 1. Now

f = \frac{∥f(n_{1})∥}{∥f(n_{1})∥ + ∥f(n_{2})∥} g + \frac{∥f(n_{2})∥}{∥f(n_{1})∥ + ∥f(n_{2})∥} h = tg + (1 − t)h, 0 < t < 1,

where t = \frac{∥f(n_{1})∥}{∥f(n_{1})∥+∥f(n_{2})∥}.

Hence f is not an extreme point. Thus f must be of the form

f(n) = δ_{nn_{0}} · x_{0},

where δ_{ij} stands for the Kronecker’s delta.

Now we claim that x_{0} is an extreme point of B_{1}(X). Indeed, ∥f∥_{ϕ} = 1 = ϕ(∥x_{0}∥). Since ϕ is strictly increasing, we have ∥x_{0}∥ = 1. If x_{0} is not an extreme point, then x_{0} = \frac{1}{2}(y + z) for some y and z in B_{1}(X). Then one can construct f_{1} and f_{2} in B_{1}(ℓ^{ϕ}(X)) such that f = \frac{1}{2}(f_{1} + f_{2}). Hence x_{0} must be extreme.

Conversely: (ii)⟶(i). Let f(n) = δ_{nn_{0}} ·x with x an extreme point of B_{1}(X). If f is not extreme, then there exist g and h in B_{1}(ℓ^{ϕ}(X)) such that f = \frac{1}{2}(g+h). But then g(n_{0}) = h(n_{0}) = x since x is an extreme point. Since ∥x∥ = 1 and ϕ is strictly increasing and ϕ(1) = 1, we have g(n) = h(n) = 0 for all n ≠ n_{0}. But this implies that f = g = h, and f is extreme. This ends the proof of the theorem.

As a corollary, we get

Theorem 2.2. A point x is an extreme point of B_{1}(ℓ^{ϕ}) if and only if x_{n} = 0 for all n except for one n, say, n_{0}, and |x_{n_{0}}| = 1.

Proof. Take R for X.

As for the exposed points we have

Theorem 2.3. Let f ∈ B_{1}(ℓ^{ϕ}(X)). The following statements are equivalent: (i) f is an exposed point. (ii) f(n) = δ_{nn_{0}} · x and x is an exposed point of B_{1}(X).

Proof. (i)⟶(ii). Let f be exposed. Then f is an extreme point. Hence f(n)δ_{nn_{0}} · x with x an extreme point of B_{1}(X). If x is not exposed, then for every x^{∗} ∈ B_{1}(X^{∗}) with x^{∗}(x) = 1, there exists z ∈ B_{1}(X) such that x^{∗}(z) = 1 and z ≠ x. Now let F ∈ [ℓ^{ϕ}(X)]^{∗} = ℓ^{∞}(X^{∗}) such that ∥F ∥ = 1, and F (f) = 1. In that case, if F = (x_{1}^{∗}, x_{2}^{∗}, …), then F (f) = x_{n_{0}}^{∗}(x) = 1. Since x is not exposed, there exists z ≠ x in B_{1}(X) such that x_{n_{0}}^{∗}(z) = 1. But then F (g) = 1, where g(n) = δ_{nn_{0}} · z and f is not exposed. Hence x must be exposed in B_{1}(X).

Conversely: (ii)⟶(i). Let f = δ_{nn_{0}} · x with x exposed in B_{1}(X). If x^{∗} is the functional that exposes x, then one can easily see that F (n) = δ_{nn_{0}} · x^{∗} is the functional that exposes f. This ends the proof.

Theorem 2.3 readily implies

Theorem 2.4. An element f is an exposed point of B_{1}(ℓ^{ϕ}) if and only if f is extreme.

As for smooth points we have

Theorem 2.5. B_{1}(ℓ^{ϕ}(X)) has no smooth points for any Banach space X.

Proof. Let f ∈ B_{1}(ℓ^{ϕ}(X)). If there exists F ∈ B_{1}(ℓ^{∞}(X^{∗})) such that F (f) = 1, then by Remark 1 f must have only one nonzero coordinate, say, f(n_{0}) = x_{n_{0}}. Since ϕ(1) = 1, it follows that ∥x_{n_{0}}∥ = 1. Consider the functionals:

F_{1}(n) = δ_{nn_{0}} · x^{∗}with x^{∗}(x_{n_{0}}) = 1,
F_{2}(n) = δ_{nn_{0}} · x^{∗} + δ_{n,n_{0}+1} · z^{∗} with ∥z^{∗}∥ = 1.

Then, F_{1} and F_{2} are two different elements in B_{2}(ℓ^{ϕ}(X)) such that F_{1}(f) = F_{2}(f) = 1. Thus f is not smooth. This ends the proof.

It follows that B_{1}(ℓ^{ϕ}) has no smooth points.

3. Geometry of B_{1}(L(ℓ^{p})), 0 < p < 1. The characterization of the extreme points of B_{1}(L(ℓ^{p})), 1 < p < ∞, is still an open difficult problem [1], [3]. In this section we give a complete description of the extreme points and the exposed points of the unit ball of L(ℓ^{p}) for 0 < p < 1. We remark that Kalton, [2], studied isomorphisms of and some classes of operators on ℓ^{p}, 0 < p < 1.

Theorem 3.1. Let T ∈ B_{1}(L(ℓ^{p})), 0 < p < 1. The following statements are equivalent:

(i) T is an extreme point.

(ii) T is a permutation on the basis elements.

Proof. (ii)⟶(i). Let T be a permutation of the basis elements e_{1}, e_{2}, . . .. If T is not extreme, then there exists S ∈ B_{1}(L(ℓ^{p})) such that S ≠ 0 and ∥S ± T ∥ ≤ 1. Thus ∥(S ± T)x∥ ≤ 1 for all x in B_{1}(ℓ^{p}). Thus, in particular, ∥Se_{n} ± T e_{n}∥ ≤ 1 for all n. Since ∥S∥ ≤ 1, it follows that T e_{n} is not extreme for those n for which Se_{n} ≠ 0. Since S ≠ 0, we get a contradiction, noting that ±e_{n} are the extreme points of ℓ^{p}. Thus T must be extreme.

Conversely: (i)⟶(ii). Let T be an extreme element of B_{1}(L(ℓ^{p})), but, if it is possible, assume there exists k_{0} such that T e_{k_{0}} is not a basis element and hence not an extreme element of B_{1}(ℓ^{p}). Thus there exists z in B_{1}(ℓ^{p}) such that ∥T e_{k_{0}} ± z∥ ≤ 1. Define the operator S on ℓ^{p} as S = e_{k_{0}} ⊗ z, so Sx = x_{k_{0}}z. Then

∥(S ± T)x∥_{p} = ∥(S ± T)(∑ x_{i}e_{i})∥_{p} = ∥∑ x_{i}(S ± T)e_{i}∥_{p}
≤ ∑ |x_{i}|^{p} ∥(S ± T)e_{i}∥_{p} .

But

(S ± T)e_{i} = \begin{cases}T e_{0}, & i ≠ k_{0}, \\ z ± T e_{k_{0}}, & i = k_{0}. \end{cases}

Thus in either case we have ∥(S ± T)e_{i}∥ ≤ 1 for all i. So ∥(S ± T)x∥ ≤ ∑ |x_{i}|^{p}. It follows that ∥S ± T ∥ ≤ 1, and T is not extreme, which contradicts the assumption. So T must be a permutation. This ends the proof.

To characterize the exposed points, we need

Theorem 3.2. L(ℓ^{p}) is isometrically isomorphic to ℓ^{∞}(ℓ^{p}).

Proof. Let f ∈ ℓ^{∞}(ℓ^{p}). Then f : N ⟶ ℓ^{p} with sup_{n} ∥f(n)∥_{p} < ∞. Define T :

ℓ^{p} ⟶ ℓ^{p}, by T x = ∑ x_{k}f(k). Then ∥T x∥_{p} ≤ ∑ ∥x_{p}f(k)∥_{p} ≤ ∑ |x_{k}|^{p} ∥f(k)∥_{p} ≤ ∥f∥_{∞} ∥x∥_{p}. Thus ∥T ∥ ≤ ∥f∥_{∞}. But T e_{k} = f(k). So ∥f(k)∥_{p} = ∥T e_{k}∥_{p} ≤ ∥T ∥. It follows that ∥f∥_{∞} ≤ ∥T ∥ .Hence ∥f∥_{∞}=∥T ∥ .

On the other hand, let T ∈ L(ℓ^{p}). Define f(n) = T e_{p}. Then one can easily show that f ∈ ℓ^{∞}(ℓ^{p}) and ∥f∥_{∞}=∥T ∥. This ends the proof.

Now for the exposed points we have

Theorem 3.3. Let T ∈ B_{1}(L(ℓ^{p})). The following statements are equivalent: (i) T is exposed. (ii) T is extreme.

Proof. That (i)⟶(ii) is immediate.

For the converse, let T be an extreme point. By Theorem 3.1, T is a permutation of the basis elements. Let f be the function corresponding to T as in Theorem 3.2. Thus f(n) = ±e_{k(n)}. Define G : L(ℓ^{p}) ⟶ R, G(S) = ∑ t_{n}⟨f(n), g(n)⟩, where 0 < t_{n}, ∑ t_{n} = 1, and g is the element in ℓ^{∞}(ℓ^{p}) that represents S as in Theorem 3.2. Then, G is bounded and ∥G∥ ≤ 1. Further G(T) = 1. Now, if it is possible, assume there exists some S in B_{1}(L(ℓ^{p})) such that G(S) = 1. Then ∑ t_{n}⟨f(n), g(n)⟩ = 1. This implies that ⟨f(n), g(n)⟩ = 1. Since f(n) = e_{k(n)}, it follows that g(n) = f(n), and so S = T. Hence T is exposed. This ends the proof.

Acknowledgment

The authors would like to thank the referee for a very careful reading of the manuscript. His suggestions and comments helped us to write the final version of the paper.

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(Received 9.02.2001; revised 4.10.2001)

Authors’ address: Department of Mathematics University of Jordan, Amman Jordan E-mail: roshdi@ju.edu.jo

hussein@ju.edu.jo wamin@ju.edu.jo