OPTIMAL AND ONE-COMPLEMENTED SUBSPACES

Grzegorz Lewicki and Giulio Trombetta

IMUJ Preprint 2005/12

Abstract. Let X be a real Banach space and let V ⊂ X be a closed linear subspace. In [3, Prop. 5] it has been proven that if X is strictly convex, reflexive and smooth and V is an optimal subset of X then V is one-complemented in X. In this note we would like to extend this result to non-smooth Banach spaces. In particular, we show that any existence subspace of c, c_{o} and l_{1} is one-complemented. Also some results concerning non-smooth Musielak-Orlicz sequence spaces equipped with the Luxemburg norm will be presented.

AMS Classification: 47B37, 46E30, 47H09.

0. Introduction.

Let X be a Banach space and let C ⊂ X be a non-empty set. A continuous mapping P : X → C is called a projection onto C, whenever P |_{C} = Id, that is P^{2} = P. Setting

Min(C) = {z ∈ X : for every c ∈ C, x ∈ X, if ∥z − c∥ ≥ ∥x − c∥ then x = z},

we say that C ⊂ X is optimal if Min(C) = C. Observe that for any C ⊂ X, C ⊂ Min(C). This notion has been introduced by Beauzamy and Maurey in [3], where basic properties concerning optimal sets can be found.

A set C ⊂ X is called an existence set of best coapproximation (existence set for brevity), if for any x ∈ X, R_{C}(x) ≠ ∅, where

R_{C}(x) = {d ∈ C : ∥d − c∥ ≤ ∥x − c∥ for any c ∈ C}.

It is clear that any existence set is an optimal set. The converse, in general, is not true. However, by [3, Prop. 2] if X is one-complemented in X^{∗∗} and strictly

Typeset by AMS-TEX

convex, then any optimal subset of X is an existence set in X, which, in particular, holds true for strictly convex spaces X, such that X = Z^{∗} for some Banach space Z. Existence and optimal sets have been studied by many authors from different points of view, mainly in the context of approximation theory (see e.g. [1 - 4], [6 - 13], [18], [24], [26], [27]). Recall that a closed subspace V of a Banach space X is called one-complemented if there exists a linear projection of norm one from X onto V. Also there is a large number of papers concerning one-complemented subspaces (see e.g. a survey paper [25] and a recent paper [16]). It is obvious that any one-complemented subspace is an existence set. The converse, in general, is not true. By a deep result of Lindenstrauss, [20] there exists a Banach space X and V a linear subspace of X, codim(V) = 2, such that:

a). V is one-complemented in any containing it hyperplane Y of X; b). V is not one-complemented in X.

This gives you immediately an example of a subspace being an existence set which is not one-complemented. However, in [3] (see also [26, p.132], where the case of contractive retractions onto linear subspaces is considered) the the following result has been proven.

Theorem 0.1. (see [3, Prop. 5]). Let V be a linear subspace of a smooth, reflexive and strictly convex Banach space. If V is an optimal set then V is onecomplemented in X. If X is a smooth Banach space, then any subspace of X which is an existence set is one-complemented. Moreover, in both cases a norm-one projection from X onto V is uniquely determined.

The aim of this paper is to generalize the above result to the case of some nonsmooth, real Banach spaces. This can be treated as a partial answer concerning the question from [3, p.125] about generalizations of [3, Prop. 5] to the non-smooth case. In particular, we show that in c, c_{o} and l_{1} any subspace which is an existence set is one-complemented (see Theorems 2.2, 2.3 and 2.4). Next we demonstrate some results in the case of non-smooth Musielak-Orlicz sequence spaces equipped with the Luxemburg norm. Now we present some notions and results which will be used in this paper. In the sequel by S(X) we denote the unit sphere in a Banach space X and by S(X^{∗}) the unit sphere in its dual space. A functional f ∈ S(X^{∗}) is called a supporting functional for x ∈ X, if f(x) = ∥x∥. Analogously, a point x ∈ S(X) is called a norming point for f ∈ X^{∗} if f(x) = ∥f∥. A point x ∈ X is called a smooth point if it has exactly one supporting functional. A Banach space X is called smooth if any x ∈ S(X) is a smooth point. By ext(X) we denote the set of all extreme points of S(X). A Banach space X is called strictly convex if ext(X) = S(X). If V is a linear subspace of a Banach space X, by P(X, V) we will denote the set of all linear, continuous projections from X onto V. Now we present some introductory facts on Musielak-Orlicz spaces. A function ϕ : ℝ → [0, +∞) is said to be an Orlicz function if ϕ(0) = 0, ϕ is even, strictly increasing and convex. By ϕ^{∗} we denote its conjugate function in the sense of Young, that is

ϕ^{∗}(u) = sup_{v>0}{|u|v − ϕ(v)},

for u ∈ ℝ and we notice that ϕ^{∗} is an extended real-valued convex function. If ϕ(u) = (1/p)u^{p}, 1 < p < ∞, then ϕ^{∗}(u) = (1/p′)u^{p′}, where 1/p+1/p′ = 1. Further, a sequence Φ = (ϕ_{n}) of Orlicz functions ϕ_{n} will be called a Musielak-Orlicz function whenever ϕ_{n}(1) = 1 for every n ∈ ℕ. By Φ^{∗} = (ϕ_{n}^{∗}) we will denote its conjugate function.

Let l_{0} denote the space of all real-valued sequences. With each Musielak-Orlicz function Φ we can associate a mapping ρ_{Φ} : l_{0} → [0, +∞] defined by

ρ_{Φ}(x) = ∑_{n=1}^{∞} ϕ_{n}(|x_{n}|),

where x = (x_{n}) ∈ l_{0}. Given a Musielak-Orlicz function Φ, let l_{Φ} denote the corresponding Musielak-Orlicz space, that is

(0.1)  l_{Φ} = {x ∈ l_{0} : lim_{λ→0} ρ_{Φ}(λx) = 0}.

If a sequence Φ = (ϕ_{n}) is constant, that is ϕ_{n} = ϕ for every n ∈ ℕ, then l_{Φ} is an Orlicz sequence space and further it will be denoted by l_{ϕ}. The space l_{Φ} equipped with the Luxemburg norm

(0.2)  ∥x∥ = ∥x∥_{Φ} := inf{λ > 0 : ρ_{Φ}(x/λ) ≤ 1}

is a Banach space.

Observe that the assumption ϕ_{n}(1) = 1 for every n ∈ ℕ is not a real restriction on Musielak-Orlicz function Φ. In fact, for every sequence Φ = (ϕ_{n}), where ϕ_{n} are Orlicz functions, there exists a function Ψ = (ψ_{n}) with ψ_{n}(1) = 1 and such that l_{Φ} is isometric to l_{Ψ}. It is enough to take ψ_{n}(t) = ϕ_{n}(a_{n}t), where ϕ_{n}(a_{n}) = 1 for every n ∈ ℕ.

We will also consider here the finite dimensional spaces l_{Φ}^{(m)}, defined on ℝ^{m} analogously as l_{Φ}. The space l_{Φ}^{(m)} can be identified with the subspace of l_{Φ} consisting of all x = (x_{n}) ∈ l_{Φ} such that x_{n} = 0 for all n ≥ m + 1.

An important subspace of l_{Φ}, called the subspace of finite elements and denoted by h_{Φ} is defined as

(0.3)  h_{Φ} = {x ∈ l_{Φ} : ρ_{Φ}(λx) < ∞ for any λ > 0}.

It is well known that h_{Φ} is a closed separable subspace of l_{Φ} with the Schauder basis consisting of the standard unit vectors e_{i} = (0, …, 1_{i}, 0, …). It is easy to see that for every x ∈ h_{Φ}, ∥x∥ = 1 if and only if ρ_{Φ}(x) = 1. Moreover, h_{Φ} = l_{Φ} if and only if either the dimension of l_{Φ} is finite or Φ satisfies a growth condition called δ_{2} [19, 22], that is there exist K, δ > 0 and a nonnegative sequence (c_{n}) ⊂ l_{1} such that for every n ∈ ℕ and every t ≥ 0

(0.4)  ϕ_{n}(2t) ≤ Kϕ_{n}(t) + c_{n},

whenever ϕ_{n}(t) ≤ δ.

Recall that for every y ∈ l_{Φ^{∗}}, the functional

f_{y}(x) = ∑_{n=1}^{∞} x_{n}y_{n}, x = (x_{n}) ∈ l_{Φ},

is bounded on (l_{Φ}, ∥ ∥_{Φ}) and is called a regular functional. We denote by R_{Φ} the set of all regular functionals on l_{Φ}. The spaces R_{Φ} and l_{Φ^{∗}} are order isomorphic [14] and so by usual identification we often write f_{y} = y. We say that f_{S} is a singular functional on l_{Φ} whenever f_{S}(x) = 0 for every x ∈ h_{Φ}. The set of all singular functionals on l_{Φ} will be denoted by S_{Φ}. It is well known [14, Lemma 1.1 and Theorem 2.9] that

(l_{Φ})^{∗} = R_{Φ} ⊕_{1} S_{Φ}.

More precisely, for each f ∈ (l_{Φ})^{∗} there exist uniquely determined r(f) ∈ R_{Φ} and s(f) ∈ S_{Φ} such that

f = r(f) + s(f),

and

∥f∥ = ∥r(f)∥ + ∥s(f)∥.

More information on Musielak-Orlicz spaces one can find in [5], [19], [22], [23], [28].

The following description of supporting functionals can be deduced from [14, Lemma 1.7 and Theorem 1.9]. Set for any i ∈ ℕ, and x ∈ l_{Φ}

ΔΦ_{i}(x) = [ϕ_{i}^{−}(x_{i}), ϕ_{i}^{+}(x_{i})],

if x_{i} ≥ 0 and

ΔΦ_{i}(x) = [ϕ_{i}^{+}(x_{i}), ϕ_{i}^{−}(x_{i})],

if x_{i} < 0. Here for any i ∈ ℕ and x ∈ ℝ ϕ_{i}^{+}(x) (ϕ_{i}^{−}(x), resp.) denote the right-hand side (the left hand-side, resp.) derivative of ϕ_{i} at x.

Theorem 0.2. Let Φ = (ϕ_{n}) be a Musielak-Orlicz function and let x = (x_{n}) ∈ h_{Φ}, ∥x∥_{Φ} = 1. Assume that (ϕ_{n})′(0) = 0 for any n ∈ ℕ. Set

A_{x} = {i ∈ ℕ : x_{i} ≠ 0}.

Then any supporting functional f_{x} of x in l_{Φ} (with respect to ∥ · ∥_{Φ}) is a regular functional. Moreover, for any y = (y_{n}) ∈ l_{Φ}

f_{x}(y) = ( ∑_{i∈A_{x}} d_{i}y_{i})/( ∑_{i∈A_{x}} d_{i}x_{i}),

where d_{i} ∈ ΔΦ_{i}(x) for any i ∈ A_{x} and ∑_{i∈A_{x}} d_{i}x_{i} < ∞.

From [14, Th.3.1] we can easily obtain

Theorem 0.3. (l_{Φ}, ∥ · ∥_{Φ}) is smooth if and only if Φ satisfies δ_{2} condition and for any i ∈ ℕ ϕ_{i} is differentiable in [0,1).

Also we need a well known

Theorem 0.4. (l_{Φ}, ∥ · ∥_{Φ}) is reflexive if and only if Φ and Φ^{∗} satisfy δ_{2} condition, that is if and only if l_{Φ} = h_{Φ} and l_{Φ^{∗}} = h_{Φ^{∗}}.

In the sequel we will apply the following results.

Theorem 0.5. [15, 17] (h_{Φ}, ∥ · ∥_{Φ}) or (l_{Φ}^{(m)}, ∥ · ∥_{Φ}) is strictly convex if and only if all functions ϕ_{n} but one are strictly convex in [0, 1). An Orlicz space (h_{ϕ}, ∥ · ∥_{Φ}) or (l_{ϕ}^{(m)}, ∥ · ∥_{ϕ}) for m ≥ 3 is strictly convex if and only if ϕ is strictly convex on the interval [0, u_{o}], where ϕ(u_{o}) = 1/2.

Theorem 0.6. (The S̆mulian Theorem) Assume X is a Banach space and let x ∈ S(X) be a smooth point. If f_{n}, g_{n} ∈ S(X^{∗}) are such that f_{n}(x) → 1 and g_{n}(x) → 1, then f_{n} − g_{n} → 0 weakly^{∗} in X^{∗}.

Theorem 0.7. (The Mazur Theorem) Let X be a separable Banach space. Then the set of all smooth points of X is a dense G_{δ} subset of X.

1. General results.

We start from

Lemma 1.1. Let X be a Banach space. Let for n ∈ ℕ x_{n} ∈ X and let ∥x_{n}−x∥ → 0 for some x ∈ X. For each n ∈ ℕ fix f_{n} ∈ S(X^{∗}) with f_{n}(x_{n}) = ∥x_{n}∥. Set for n ∈ ℕ

A_{n} = cl({f_{k} : k ≥ n}),

where the closure is taken with respect to the weak-∗ topology in X^{∗}. Then for any f ∈ A({f_{n}}) = ⋂_{n=1}^{∞} A_{n}, f(x) = ∥x∥ and ∥f∥ = 1, if x ≠ 0.

Proof. If x = 0, then obviously f(x) = ∥x∥ for any f ∈ A({f_{n}}). If x ≠ 0, then x_{n}/∥x_{n}∥ → x/∥x∥. Hence without loss of generality we can assume that ∥x_{n}∥ = ∥x∥ = 1. Let f ∈ A({f_{n}}). By the Banach-Alaoglu Theorem ∥f∥ ≤ 1. Fix ϵ > 0 and n_{o} ∈ ℕ with ∥x_{n} − x∥ ≤ ϵ/2 for n ≥ n_{o}. Also there exists k ≥ n_{o} with |(f_{k} − f)x| ≤ ϵ/2. Note that

|1 − f(x)| ≤ |f_{k}(x_{k} − x)| + |(f_{k} − f)x|
≤ ∥f_{k}∥∥x_{k} − x∥ + |(f_{k} − f)x| ≤ ϵ.

Consequently, f(x) = ∥x∥ = 1 = ∥f∥, as required.

Lemma 1.2. Let V ⊂ X be a linear subspace of a Banach space X. Take x, y ∈ X satisfying

∥y − v∥ ≤ ∥x − v∥

for any v ∈ V. Then for any v ∈ V there exists f_{v} ∈ S(X^{∗}), f_{v}(v) = ∥v∥ and f_{v}(x − y) = 0.

Proof. Fix v ∈ V. Without loss of generality, we can assume that v ≠ 0. Since V is a linear subspace, for any k ∈ ℝ

∥y − kv∥ ≤ ∥x − kv∥.

Hence for any k ≠ 0,

∥y/k − v∥ ≤ ∥x/k − v∥.

Let k ∈ ℕ \ {0}. Choose f_{k} ∈ S(X^{∗}) such that

f_{k}(x/k − v) = ∥x/k − v∥.

Note that

f_{k}(y/k − v) ≤ ∥y/k − v∥ ≤ ∥x/k − v∥.

Hence f_{k}(x/k − y/k) ≥ 0, which gives

f_{k}(y − x) ≤ 0

for any k ∈ ℕ \ {0}. Observe that ∥x/k − v − (−v)∥ → 0. By Lemma 1.1 and the Banach - Alaoglu Theorem there exists f_{+} ∈ A({f_{k}}) such that f_{+}(−v) = ∥v∥. Since f_{+} ∈ A({f_{k}}), f_{+}(y − x) ≤ 0. For k ∈ ℤ \ ℕ, by the above reasoning, f_{k}(x/k − y/k) ≥ 0, and consequently

f_{k}(y − x) ≥ 0.

Again, by Lemma 1.1 and the Banach - Alaoglu Theorem, there exists f_{−} ∈ S(X^{∗}), f_{−}(−v) = ∥v∥ such that

f_{−}(y − x) ≥ 0.

If f_{−}(y − x) = 0 or f_{+}(y − x) = 0, the lemma is proved. In the opposite case, there exists a ∈ (0, 1) such that

af_{−}(x − y) + (1 − a)f_{+}(x − y) = 0

Set

f_{v} = −(af_{−} + (1 − a)f_{+}).

Obviously, ∥f_{v}∥ = f_{v}(v/∥v∥) = 1 and f_{v}(y − x) = 0. The lemma is proved.

Theorem 1.3. Let X be a real Banach space and let V ⊂ X, be a linear subspace. Assume that V is an existence set and V ≠ {0}. Put

(1.1) G_{V} = {v ∈ V \ {0} : there exists exactly one f ∈ S(X^{∗}) : f(v) = ∥v∥}.

Assume that the norm closure of G_{V} in X is equal to V. Then there exists exactly one projection P ∈ P(X, V) such that ∥P ∥ = 1.

Proof. Fix x ∈ X. Since V is an existence set, there exists y ∈ V such that

∥y − v∥ ≤ ∥x − v∥

for any v ∈ V. By Lemma 1.2 applied to x and y for any v ∈ V there exists f_{v} ∈ S(X^{∗}), f_{v}(v) = ∥v∥, such that

(1.2)  f_{v}(x − y) = 0.

Now we show that there exists exactly one y ∈ V satisfying (1.2) for any v ∈ V. Assume, on the contrary, that there exist y_{1}, y_{2} ∈ V, y_{1} ≠ y_{2} satisfying (1.2) for any v ∈ V. Since cl(G_{V}) = V, there exists {z_{n}} ⊂ G_{V} , with

(1.3)  ∥z_{n} − (y_{1} − y_{2})∥ → 0.

By (1.2), for any n ∈ ℕ there exists f_{n}^{1} ∈ S(X^{∗}) and f_{n}^{2} ∈ S(X^{∗}), with f_{n}^{i}(z_{n}) = ∥z_{n}∥ for i = 1, 2 such that f_{n}^{i}(x − y_{i}) = 0. Since z_{n} ∈ G_{V} , f_{n}^{1} = f_{n}^{2}. Hence for any n ∈ ℕ

(1.4)  f_{n}^{1}(y_{1} − y_{2}) = 0.

Since ∥z_{n} −(y_{1} −y_{2})∥ → 0, and f_{n}^{1}(z_{n}) = ∥z_{n}∥, by Lemma 1.1, for any f ∈ A({f_{n}^{1}})

(1.5)  f(y_{1} − y_{2}) = ∥y_{1} − y_{2}∥.

By (1.4), f(y_{1} − y_{2}) = 0, which gives, y_{1} = y_{2}; a contradiction. Now, let for any x ∈ X, P x denote the only element y ∈ V satisfying (1.2) for any v ∈ V. We show that P is a linear mapping. To do this, fix x_{1}, x_{2} ∈ X and v ∈ G_{V} . Note that f_{v}(x_{1} − P x_{1}) = 0 and f_{v}(x_{2} − P x_{2}) = 0, where f_{v} denote the only supporting functional for v in X^{∗}. Consequently, for any v ∈ G_{V} , a_{1}, a_{2} ∈ ℝ,

(1.6)  f_{v}(a_{1}x_{1} + a_{2}x_{2} − (a_{1}P x_{1} + a_{2}P x_{2})) = 0.

Now assume v ∈ V \ G_{V} . Since the norm closure of G_{V} is equal to V, there exists {z_{n}} ⊂ G_{V} , ∥z_{n} − v∥ → 0. By Lemma 1.1, the Banach - Alaoglu Theorem and (1.6), there exists f ∈ S(X^{∗}), f(v) = ∥v∥, such that

(1.7)  f(a_{1}x_{1} + a_{2}x_{2} − (a_{1}P x_{1} + a_{2}P x_{2})) = 0.

Since for any x ∈ X there exists exactly one element satisfying (1.2) for any v ∈ V, by (1.6) and (1.7),

P (a_{1}x_{1} + a_{2}x_{2}) = a_{1}P (x_{1}) + a_{2}P (x_{2}),

which shows that P is a linear mapping. Taking v = 0 we get ∥P x∥ ≤ ∥x∥. Since P v = v for any v ∈ V and V ≠ {0}, we get ∥P ∥ = 1. By the above proof there exists exactly one projection P ∈ P(X, V) of norm one. The proof is complete.

Corollary 1.4. Assume X is a smooth space. If a linear subspace V ⊂ X, V ≠ {0}, is an existence set then V is one-complemented in X.

Proof. Note that in our case G_{V} = V \{0}. Hence the statement follows immediately from Theorem 1.3.

Corollary 1.5. Assume that X is a strictly convex Banach space such that X is one-complemented in X^{∗∗}. (In particular, X can be a reflexive space or by X = Z^{∗} for some Banach space Z.) If a linear subspace V ⊂ X, V ≠ {0}, is an optimal set satisfying the requirements of Theorem 1.3, then V is one-complemented.

Proof. By [3, Prop. 2] V is an existence set in X. By Theorem 1.3, V is onecomplemented.

Now we present another class of subspaces satisfying the assumptions of Theorem 1.3.

Definition 1.6. Let X be a Banach space and let V ⊂ X be a linear subspace. V is called weakly separating if any g ∈ ext(V^{∗}) has exactly one Hahn-Banach extension f ∈ S(X^{∗}).

Theorem 1.7. Let V ⊂ X, V ≠ {0} be a separable, weakly separating subspace. Then the norm closure of G_{V} is equal to V.

Proof. Since V is separable, by Theorem 0.7, the set

(1.8) G_{V,1} = {v ∈ V \ {0} : there exists exactly one g ∈ S(V^{∗}), g(v) = ∥v∥}

is dense in V. To finish the proof, it is enough to show that G_{V,1} = G_{V} . Fix v ∈ G_{V,1} and g ∈ S(X^{∗}) satisfying g(v) = ∥v∥. Since v ∈ G_{V,1}, g ∈ ext(V^{∗}). Now we show that there exists exactly one f ∈ S(X^{∗}) such that f(v) = ∥v∥, which means that v ∈ G_{V} . Indeed, assume there exist f_{1}, f_{2} ∈ S(X^{∗}), f_{1} ≠ f_{2} such that f_{i}(v) = ∥v∥ for i = 1, 2. Hence f_{i}|_{V} (v) = ∥v∥ for i = 1, 2. Since v ∈ G_{V,1}, f_{1}|_{V} = f_{2}|_{V} = g. Since g ∈ ext(V^{∗}), and V is weakly separating, f_{1} = f_{2}; a contradiction. It is obvious, that G_{V} ⊂ G_{V,1}. The proof is complete.

Now we present some examples of weakly separating subspaces of C(E), where E is a compact set and C(E) is the space of continuous, real-valued functions defined on E equipped with the supremum norm.

Theorem 1.8. [21] A linear subspace V ⊂ C(E), is weakly separating if and only if for any t_{1}, t_{2} ∈ A_{V} , t_{1} ≠ t_{2}, there exist v_{1}, v_{2} ∈ V such that

v_{1}(t_{1}) ≠ v_{1}(t_{2})

and

v_{2}(t_{1}) ≠ −v_{2}(t_{2}).

Here

C_{V} = {t ∈ E : t̂|_{V} ∈ ext(V^{∗})}.

Example 1.9. Let V ⊂ C(E) be a linear subspace such that 1 ∈ V and t_{1}|_{V} ≠ t_{2}|_{V} for any t_{1}, t_{2} ∈ E. Then V is weakly separating.

Example 1.10. Let V ⊂ C(E) be a Haar subspace of dimension ≥ 2. Then V is weakly separating.

2. Particular cases.

Now applying Theorem 1.3, we show that in c and c_{o} any linear subspace V ≠ {0}, which is an existence set must be one-complemented. First we need a well known

Lemma 2.1. Let X and Y be two Banach spaces and let T : X → Y be a linear isometry. Let V ⊂ X. Then V is an existence set in X if and only if T (V) is an existence set in T (X). If V ⊂ X, V ≠ {0}, is a linear space then V is onecomplemented in X if and only if T (V) is one-complemented in T (X).

Proof. Since V is an existence set for any x ∈ X there exists Qx ∈ V such that for any v ∈ V

∥v − Qx∥ ≤ ∥v − x∥.

Hence

∥T (v) − T (Qx)∥ ≤ ∥T (v) − T (x)∥

for any v ∈ V. This means that T (V) is an existence set in T (X). Applying the above reasoning to T (X) and T^{−1} we get the first claim of our lemma. Now suppose that V is one-complemented subspace of X. Take a projection P_{o} ∈ P(X, V), ∥P_{o}∥ = 1. Set P_{1} = T ∘ P_{o} ∘ T^{−1}. Then obviously P_{1} ∈ P(T (X), T (V)), and ∥P_{1}∥ = 1. The converse is obvious.

Theorem 2.2. Let V ⊂ c, V ≠ {0}, be a linear subspace which is an existence set. Then V is one-complemented in c.

Proof. Since by the Hahn-Banach theorem any one-dimensional subspace of c is one-complemented, we can assume that dim(V) ≥ 2. For each i ∈ ℕ set

C_{i,1} = {j ∈ ℕ, j ≠ i, v_{i} = v_{j} for any v ∈ V },
C_{i,2} = {j ∈ ℕ, j ≠ i, v_{i} = −v_{j} for any v ∈ V }

and

C_{i} = {i} ∪ C_{i,1} ∪ C_{i,2}.

Observe that for any i, j ∈ ℕ,

(2.1)  C_{i} ∩ C_{j} = ∅ or C_{i} = C_{j}.

Also it is clear that

⋃_{i=1}^{∞} C_{i} = ℕ.

First assume that for any n ∈ ℕ

(2.2)  ⋃_{i=1}^{n} C_{i} ≠ ℕ.

Define

i_{1} = 1 = min(C_{1}),
i_{2} = min(ℕ \ C_{1}),

and

i_{n} = min(ℕ \ ⋃_{j=1}^{n−1} C_{i_{j}}).

Note that by (2.2), the above definition is correct. Put

N_{1} = {n ∈ ℕ : card(C_{i_{n}}) < ∞}.

Since V ⊂ c, N_{1} = ℕ or there exists exactly one n_{o} ∈ ℕ such that

(2.3)  N_{1} = ℕ \ {n_{o}}.

Set

(2.4)  V_{1} = {x ∈ c : there exists v ∈ V, x_{j} = v_{i_{j}} for j ∈ N_{1}}.

Note that for any x ∈ V_{1} there exists exactly one v ∈ V such that (2.4) is satisfied. Let us denote it by Lx. Observe that L is a linear isometry between V_{1} and V equal to the identity mapping if C_{i} = {i} for any i ∈ ℕ.

Now we show that V_{1} is an existence set in c. First we assume that N_{1} = ℕ. Take x ∈ c. Define a sequence T x by

(2.5)  T (x)_{k} = \{ \begin{matrix}x_{j}, & if k ∈ C_{i_{j},1} ∪ \{i_{j}\}, \\ −x_{j} & if k ∈ C_{i_{j},2}. \end{matrix}

By (2.1), T (x) is properly defined. If N_{1} = ℕ \ {n_{o}} (see (2.3)), we modify our definition for k ∈ C_{i_{no} ,i}, if card(C_{i_{no} ,i}) = ∞, for i = 1 or i = 2 setting

(2.6)  T (x)_{k} = lim_{n} x_{n}.

By (2.2), (2.5) and (2.6) T is a linear isometry from c into c. Observe that T |_{V_{1}} = L. Since V is an existence set in c, V = T (V_{1}) ⊂ T (c) is an existence set in T (c). By Lemma 2.1, V_{1} = T^{−1}(V) is an existence set in c. Now we consider two cases. Case I. For any i ∈ ℕ there exists v ∈ V_{1} and w ∈ V_{1} with v_{i} ≠ lim_{n}v_{n} and w_{i} ≠ −lim_{n}w_{n}. We show that V_{1} is a weakly separating subspace of c.Note that c is isometric to C(E), where E = {0, 1/n : n ∈ ℕ} and an isometry I : c → C(E) is given by:

(Ix)(0) = lim_{n}(x) and (Ix)(1/n) = x_{n}.

Hence it is enough to show that I(V_{1}) is a weakly separating subspace of C(E). To do this, we apply Theorem 1.8. Take any t_{1}, t_{2} ∈ C_{V_{1}}, (see Th.1.8) t_{1} ≠ t_{2}. If t_{i} ≠ 0 for i = 1, 2, by the construction of V_{1}, there exists v_{1} ∈ V_{1}, such that

I(v_{1})(t_{1}) ≠ I(v_{1})(t_{2})

and v_{2} ∈ V_{1} with

I(v_{2})(t_{1}) ≠ −I(v_{2})(t_{2}).

If t_{1} = 0 or t_{2} = 0, we can find v_{1} ∈ V_{1} and v_{2} ∈ V_{1} satisfying the above conditions, because we consider the Case I. By Theorem 1.8, I(V_{1}) is a weakly separating subspace of C(E) and consequently V_{1} is a weakly separating subspace of c. By Theorems 1.3, 1.7 and Lemma 2.1 V = T (V_{1}) is one-complemented in T (c). Let P_{1} ∈ P(T (c), V) be a projection of norm one. Set

Q_{1} = P_{1} ∘ T ∘ R,

where

Rx = (x_{i_{1}}, x_{i_{2}}, ...)

for x ∈ c and i_{1}, i_{2}, ... ∈ N_{1}. It is clear that ∥Q_{1}∥ ≤ 1. Since R|_{V} = T^{−1}|_{V} = L^{−1}, Q_{1} is a norm-one projection from P(c, V), which completes our proof in this case. Case II. There exists i ∈ ℕ such that for any w ∈ V_{1} w_{i} = lim_{n} w_{n} or there exists i ∈ ℕ such that for any w ∈ V_{1} w_{i} = − lim_{n} w_{n}. By definition of V_{1}, there exists exactly one i ∈ ℕ satisfying the above condition. Without loss of generality we can assume that i = 1 and that for any w ∈ V, w_{1} = lim_{n} w_{n}. Let S : c → c be given by :

(2.7)  S(x) = (lim_{n} x_{n}, x).

Note that V_{1} ⊂ S(c). Set

V_{2} = {(v_{2}, ...) : v = (v_{1}, v_{2}, ...) ∈ V_{1}}.

It is clear that V_{1} = S(V_{2}). Since V_{1} is an existence set in c, and V_{1} ⊂ S(c), V_{1} is an existence set in S(c). By Lemma 2.1, V_{2} is an existence set in c. Reasoning as in the Case I, we get that V_{2} is a weakly separating subspace of c. By Theorems 1.3 and 1.7 there exists P_{2} : c → V_{2}, a linear projection of norm one. Define R_{1} : c → V_{1} by

R_{1}(x) = (S ∘ P_{2})(x_{2}, ...).

It is clear that ∥R_{1}∥ = 1, R_{1}(c) ⊂ V_{1} and R_{1}|_{V_{1}} = id_{V_{1}}. Hence V_{1} is onecomplemented in c. Reasoning as in the Case I, we get that V is one-complemented in c. This completes the proof of our theorem under assumption (2.2). If (2.2) is not satisfied, without loss of generality we can assume that

ℕ = ⋃_{j=1}^{n} C_{i_{j}},

where 1 = i_{1} < i_{2} <, …, < i_{n} and C_{i_{j}} ∩ C_{i_{k}} = ∅ for j ≠ k. Let T : l_{∞}^{(n)} → c be defined by (2.5). Note that V ⊂ T (l_{∞}^{(n)}). Let V_{1} = T^{−1}(V). Reasoning as in the first part of the proof, we get that V_{1} is an existence set in l_{∞}^{(n)} and V_{1} is a weakly separating subspace of l_{∞}^{(n)}. By Theorems 1.3 and 1.7 there exists a norm-one projection P_{3} ∈ P(l_{∞}^{(n)}, V_{1}). Set

Q_{2} = T ∘ P_{3} ∘ R.

Here R : c → l_{∞}^{(n)} is given by

Rx = (x_{i_{1}}, ..., x_{i_{n}}).

Since R|_{V} = T^{−1}|_{V} , Q_{2} ∈ P(c, V) and ∥Q_{2}∥ = 1. The proof is complete.

In analogous way we can prove

Theorem 2.3. Let V ⊂ c_{o}, V ≠ {0}, be a linear subspace which is an existence set. Then V is one-complemented in c_{o}.

Now we consider the case X = l_{1}.

Theorem 2.4. Let V ⊂ l_{1}, V ≠ {0}, be a linear subspace, which is an existence set. Then V is one-complemented in l_{1}.

Proof. Set for x ∈ l_{1}

(2.8)  A_{x} = {n ∈ ℕ : x_{n} = 0}

and

A_{V} = ⋃_{v∈V} A_{v}.

Without loss of generality we can assume that A_{V} = ℕ. We show that G_{V,1} = G_{V} (see (1.1) and (1.8)). Since (l_{1})^{∗} = l_{∞}, v ∈ G_{V,1} if and only if A_{v} = A_{V} = ℕ. Hence v ∈ G_{V} . By Theorems 0.7, 1.3 and 1.7, V is one-complemented in l_{1}.

Now we consider the case of Musielak-Orlicz sequence spaces equipped with the Luxemburg norm (see (0.2)). For a linear subspace V ⊂ l_{Φ} (see (0.1)) and n ∈ ℕ define I_{n} ∈ V^{∗} by

(2.9)  I_{n} = (y → y_{n})|_{V} .

We start from Lemma 2.5. Let V ⊂ h_{Φ} (see (0.3)) be a linear subspace, dim(V) ≥ 2. Let v ∈ G_{V,1}, (see (1.8)), ∥v∥ = 1 and card(A_{v}) ≥ 2. Then v ∈ G_{V,1} \ G_{V} if and only if there exists n_{o} ∈ ℕ such that I_{n_{o}} ≠ 0, card(ΔΦ_{n_{o}}(v)) > 1 and v is a norming point for a functional J_{n_{o}} = I_{n_{o}}/∥I_{n_{o}}∥.

Proof. Take v ∈ G_{V,1} \ G_{V} with ∥v∥ = 1 and card(A_{v}) ≥ 2. By Theorem 0.2, there exists n_{o} ∈ A_{v} and c_{n_{o}}, d_{n_{o}} ∈ ΔΦ_{n_{o}}(v), c_{n_{o}} ≠ d_{n_{o}}. Define for y ∈ l_{Φ}

f_{1}(y) = ( ∑_{n∈A_{v},n≠n_{o}} d_{n}y_{n} + d_{n_{o}}y_{n_{o}})/( ∑_{n∈A_{v}} d_{n}v_{n}),

and

f_{2}(y) = ( ∑_{n∈A_{v},n≠n_{o}} d_{n}y_{n} + c_{n_{o}}y_{n_{o}})/( ∑_{n∈A_{v},n≠n_{o}} d_{n}v_{n} + c_{n_{o}}v_{n_{o}}),

where d_{n} ∈ ΔΦ_{n}(v) for any n ∈ A_{v}, n ≠ n_{o}. Note that by Th.0.2, f_{1} and f_{2} are two different supporting functionals for v. Since v ∈ G_{V,1} \ G_{V} , f_{1}|_{V} = f_{2}|_{V} . We show that f_{1}|_{V} = J_{n_{o}}. If not, there exists w ∈ V such that f_{1}(w) ≠ 0 and I_{n_{o}}(w) = 0. Hence

f_{1}(w) = ( ∑_{n∈A_{v},n≠n_{o}} d_{n}w_{n})/( ∑_{n∈A_{v}} d_{n}v_{n}),

and

f_{2}(w) = ( ∑_{n∈A_{v},n≠n_{o}} d_{n}w_{n})/( ∑_{n∈A_{v},n≠n_{o}} d_{n}v_{n} + c_{n_{o}}v_{n_{o}}).

Since v_{n_{o}} ≠ 0 and c_{n_{o}} ≠ d_{n_{o}}, f_{1}(w) ≠ f_{2}(w); a contradiction. To prove the converse, assume that v ∈ G_{V,1}, ∥v∥ = 1, is a norming point for I_{n_{o}} ≠ 0 and card(ΔΦ_{n_{o}}(v)) > 1. Fix c, d ∈ ΔΦ_{n_{o}}(v), c ≠ d and d_{n} ∈ ΔΦ_{n}(v), for n ∈ A_{v}, n ≠ n_{o}. Set

f_{1}(y) = ( ∑_{n∈A_{v},n≠n_{o}} d_{n}y_{n} + dy_{n_{o}})/( ∑_{n∈A_{v},n≠n_{o}} d_{n}v_{n} + dv_{n_{o}})

and

f_{2}(y) = ( ∑_{n∈A_{v},n≠n_{o}} d_{n}y_{n} + cy_{n_{o}})/( ∑_{n∈A_{v},n≠n_{o}} d_{n}v_{n} + cv_{n_{o}}).

By Th. 0.2, f_{1}, f_{2} are supporting functionals for v in l_{Φ}. Since f_{1}(e_{n_{o}}) ≠ f_{2}(e_{n_{o}}), v ∈ G_{V,1} \ G_{V} , as required.

Theorem 2.6. Let Φ be a Musielak - Orlicz function. Let V ⊂ h_{Φ}, V ≠ {0}, be a linear subspace which is an existence set. Set

(2.10)  N_{1} = {n ∈ ℕ : I_{n} ≠ 0, card(ΔΦ_{n}(∥I_{n}∥)) > 1}

and for any n ∈ N_{1}

(2.11)  Z_{n} = {v ∈ S(V) : J_{n}(v) = 1}.

Assume that for any n ∈ N_{1}, int(Z_{n}) in S(V) is empty. If lim inf_{n} ∥I_{n}∥ > 0 or if there exist a > 0 and n_{o} ∈ ℕ such that ϕ_{n} are differentiable in [0, a], for n ≥ n_{o}, then V is one-complemented in l_{Φ}.

Proof. Note that, by the Hahn-Banach theorem, any one-dimensional subspace of l_{Φ} is one-complemented. Hence we can assume that dimV ≥ 2. We will apply Theorem 1.3. To do this, it is necessary to show that cl(G_{V}) = V. Since h_{Φ} is separable, by Th. 0.7, cl(G_{V,1}) = V. To end the proof, we demonstrate that any v ∈ G_{V,1}, can be approximated by elements from G_{V} . Fix v ∈ G_{V,1} \ G_{V} , ∥v∥ = 1. Since dim(V) ≥ 2, we can assume that card(A_{v}) ≥ 2. By Lemma 2.5, there exists n_{o} ∈ ℕ with I_{n_{o}} ≠ 0, card(ΔΦ_{n_{o}}(v)) ≥ 2 such that v is a norming point for J_{n_{o}}. Set

(2.12)  S_{k} = {w ∈ S(V) : ∥w − v∥ ≤ 1/k}.

Now we show that for any k ∈ ℕ there exists w_{k} ∈ G_{V,1} ∩ S_{k}, such that

(2.13)  w_{k} ∉ ⋃_{j=1}^{k}(Z_{j}).

Indeed, if there exists k ∈ ℕ such that (2.13) is not satisfied for any w ∈ G_{V,1} ∩ S_{k}, by Th. 0.7,

(2.14)  S_{k} = ⋃_{j=1}^{k}(Z_{j} ∩ S_{k}).

Since l_{Φ} is a Banach space, and V is closed, S_{k} is a complete, metric space (with respect to the norm topology). By the Baire Property and (2.14), int(Z_{j}) in S(V) is nonempty for some j ∈ {1, ..., k}; a contradiction. Now we show that for k sufficiently large there exists w_{k} ∈ S_{k} ∩ G_{V} satisfying (2.13). First assume that there exist a > 0 and n_{o} ∈ ℕ such that ϕ_{n} is differentiable for any n ≥ n_{o}. Fix for any k ∈ ℕ w_{k} ∈ G_{V,1} ∩ S_{k}. If there exists a subsequence {n_{k}} such that w_{n_{k}} ∉ G_{V} , by Lemma 2.1, w_{n_{k}} is a norming point for some J_{m_{k}}. By (2.13), m_{k} → +∞. Note that

|1 − J_{m_{k}}(v)| = |J_{m_{k}}(v_{n_{k}} − v)| ≤ ∥v_{n_{k}} − v∥ → 0.

By Theorem 0.6,

(2.15)  I_{m_{k}}/∥I_{m_{k}}∥ = J_{m_{k}} → J_{n_{o}} = I_{n_{o}}/∥I_{n_{o}}∥.

weakly^{∗} in V^{∗}. Consequently, J_{m_{k}}(v) → J_{n_{o}}(v) = 1. Since V ⊂ h_{Φ}, lim_{k}v_{k} = 0. By (2.15) ∥I_{m_{k}}∥ → 0. Since there exist a > 0 and n_{o} ∈ ℕ such that ϕ_{n} is differentiable for any n ≥ n_{o}, N_{1} is a finite set. Hence for k ≥ k_{o}

card(ΔΦ_{m_{k}}(∥I_{m_{k}}∥)) = 1.

By Lemma 2.1, w_{k} ∈ G_{V} for k ≥ k_{o} : a contradiction. If lim inf ∥I_{n}∥ > 0, we proceed in the same way as above. The proof is complete.

Theorem 2.7. Let V ⊂ l_{Φ}^{(m)}, V ≠ {0}, be a linear subspace which is an existence set. Assume that for any n ∈ N_{1}, (see (2.10)) int(Z_{n}) in S(V) is empty. Then V is one-complemented in l_{Φ}^{(m)}.

Proof. Goes in the same way as the proof of Theorem 2.6.

Now we present some examples in which int(Z_{n}) with respect to S(V) is empty. Recall that a convex function f : ℝ → ℝ is called strictly convex if for any x ≠ y and a ∈ (0, 1),

(2.16)  f(ax + (1 − a)y) < af(x) + (1 − a)f(y).

Theorem 2.8. Let Φ be a Musielak-Orlicz function such that ϕ_{k} are strictly convex for k > k_{o}. Set for any F ∈ S((l_{Φ})^{∗}), (l_{Φ}^{(m)} resp.)

(2.17)  Z_{F} = {x ∈ S(h_{Φ}) : F (x) = 1}.

Fix x ∈ Z_{F} . Then dim(span(Z_{F} − x)) < k_{o}.

Proof. Assume on the contrary, that dim(span(Z_{F} − x)) ≥ k_{o}. Hence there exists x_{1}, ..., x_{k_{o}+1} ∈ Z_{F} such that y_{i} = x_{i+1} − x_{1}, i = 1, ..., k_{o} are linearly independent. Note that by definition of Z_{F} ,

1 = F ( ∑_{j=1}^{k_{o}+1} x_{j}/(k_{o} + 1)) = ∥ ∑_{j=1}^{k_{o}+1} x_{j}/(k_{o} + 1))∥_{Φ} = ∑_{l=1}^{∞} ϕ_{l}( ∑_{j=1}^{k_{o}+1}(x_{j}/(k_{o} + 1))_{l})

(2.18)  ≤ (1/(k_{o} + 1)) ∑_{j=1}^{k_{o}+1}(∑_{l=1}^{∞} ϕ_{l}((x_{j})_{l}) ≤ (1/(k_{o} + 1)) ∑_{j=1}^{k_{o}+1} ρ_{Φ}(x_{j}) = 1.

Since for k > k_{o} ϕ_{k} are strictly convex, by (2.18), (x_{j})_{l} = (x_{1})_{l} for l > k_{o} and j = 1, ..., k_{o} + 1. Hence for j = 1, ..., k_{o},

y_{j} = x_{j+1} − x_{1} = ((x_{j} − x_{1})_{1}, ..., (x_{j} − x_{1})_{k_{o}}, 0, ...,).

Since y_{j}, j = 1, ..., k_{o} are linearly independent,

(2.19)  det((y_{j})_{k})_{j,k=1,...,k_{o}} ≠ 0.

Because of (2.18), for any l = 1, ..., k_{o} ϕ_{l} is linear in some interval E_{l} = [c_{l}, d_{l}], c_{l} < d_{l} containing all the coordinates (x_{j})_{l}, j = 1, ..., k_{o} + 1. Assume that for l = 1, ..., k_{o} + 1 and x ∈ E_{l},

ϕ_{l}(x) = a_{l}x + b_{l}.

By (2.18) for any j = 1, ..., k_{o} + 1,

(2.20)  ∑_{l=1}^{k_{o}} a_{l}(x_{j})_{l} + b_{l} = 1 − ∑_{l=k_{o}+1}^{∞} ϕ_{l}((x_{1})_{l}).

Since all the functions ϕ_{l} are even and strictly increasing in [0, +∞), a_{l} ≠ 0 for l = 1, ..., k_{o}. By (2.20) we get for j = 1, ..., k_{o},

∑_{l=1}^{k_{o}} a_{l}(y_{j})_{l} = 0,

which contradicts (2.19). The proof is complete.

From Theorem 2.8 we immediately get Corollary 2.9. Let Φ satisfy the assumptions of Theorem 2.8 with some k_{o} ≥ 1. Then for any subspace V ⊂ h_{Φ} dim(V) ≥ k_{o} and for any n ∈ ℕ int(Z_{n}) in S(V) is empty. In particular if k_{o} ≤ 2 and Φ satisfy the assumptions of Theorem 2.6 any subspace of h_{Φ} which is an existence set is one-complemented in l_{Φ}.

Corollary 2.10. Let k_{o} ≤ 2 and let V ⊂ l_{Φ}^{(m)}, V ≠ {0}, be a linear subspace which is an existence set. Then V is one-complemented. Moreover, there exists exactly one projection of norm one from l_{Φ}^{(m)}, onto V. If l_{Φ}^{(m)}, is strictly convex (see Th. 0.5) then any subspace of l_{Φ}^{(m)}, which is an optimal set is one-complemented.

Proof. Follows immediately from [3, Prop.2] Theorems 2.7 and 2.8.

Corollary 2.11. Let k_{o} ≤ 2 and let V ⊂ h_{Φ}, V ≠ {0}, be a linear subspace which is an existence set. If Φ satisfies the requirements of Theorem 2.6, then V is onecomplemented. Moreover, there exists exactly one projection of norm one from l_{Φ}, onto V. If l_{Φ}, is strictly convex (see Th. 0.5) and reflexive (see Th. 0.4), then any subspace of l_{Φ} which is an optimal set is one-complemented.

Proof. By Theorem 0.4 if l_{Φ} is reflexive then l_{Φ} = h_{Φ}. Then [3, Prop.2], Theorems 2.6 and 2.8 should be applied.

Remark 2.12. Note that c_{o}, c and l_{1} are not smooth spaces. In general, Musielak-Orlicz or Orlicz spaces satisfying the requirements of Theorem 2.6 and Th.2.7 are also not smooth (compare with Th. 0.3). This shows that Theorem 1.3, which proof is very similar to that of [3, Prop. 5], can be applied to to non-smooth case in many concrete situations.

References.

[1] B. Beauzamy, Projections contractantes dans les espaces de Banach, Bull. Sci. Math. (2) 102 (1978), 43 – 47.

[2] B. Beauzamy and P. Enflo, Théorèmes de point fixe et d’approximation, Ark. Mat. 23,1 (1985), 19 – 34.

[3] B. Beauzamy and B. Maurey, Points minimaux et ensembles optimaux dans les espaces de Banach, J. Funct. Anal. 24 (1977), 107 – 139.

[4] H. Berens, Über die beste Approximation in ℝ^{n}, Arch. Math. 39 (1982) 376 – 382.

[5] S. Bian, H.Hudzik and T. Wang, Smooth, very smooth and strongly smooth points in Musielak-Orlicz sequence spaces, Bull. Austral. Math. Soc. 63 (2001), 441 – 457.

[6] R. E. Bruck Jr., Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc. 179 (1973) 251 – 262.

[7] R. E. Bruck Jr., Nonexpansive projections on subsets of Banach spaces, Pacific J. Math. 47,2 (1973) 341 – 355.

[8] V. Davis, P. Enflo, Contractive projections on l_{p}-spaces, London Math. Soc. Lecture Notes Series 137 (1989) 151 – 161.

[9] P. Enflo, Contractive projections onto subsets of L^{1}(0, 1), London Math. Soc. Lecture Notes Series, 137 (1989) 162 – 184.

[10] P. Enflo, Contractive projections onto subsets of L^{p}-spaces, in: Lecture Notes in Pure and Applied Mathematics, Function Spaces, 136, 79 – 94, New York, Basel, Marcel Dekker Inc., 1992.

[11] D.G. de Figueiredo and L. A. Karlovitz, On the extensions of contractions of normed spaces, in: Nonlinear Functional Analysis, Proceedings of Symposia in Pure Mathematics 18,1 95 – 104, Providence 1970, Amer. Math. Soc..

[12] P. M. Gruber, Fixpunktmengen von Kontraktionen in endlichdimensionalen normierten Räumen, Geom. Dedicata 4 (1975), 179 – 198.

[13] L. Hetzelt, On suns and cosuns in finite dimensional normed real vector spaces, Math. Hungar. 45 (1985) 53 – 68.

[14] H. Hudzik, Y. Ye, Support functionals and smoothness in Musielak-Orlicz sequence spaces endowed with the Luxemburg norm, Comment. Math. Univ. Carolinae 31,4 (1990) 661 -684.

[15] H. Hudzik, D. Pallaschke On some convexity properties of Orlicz sequence spaces equipped with the Luxemburg norm, Math. Nachr. (1997) 167 – 185.

[16] J. Jamison, A. Kamińska and G. Lewicki, One-complemented subspaces of Musielak-Orlicz sequence spaces, Journ. Approx. Theory, 130 (2004) 1 – 37.

[17] A. Kamińska, Rotundity of Orlicz-Musielak sequence spaces, Bull. Acad. Polon. Sci. Sr. Sci. Math. 29 (1981), no. 3-4, 137–144.

[18] A. Kamińska, G. Lewicki, Contractive and optimal sets in modular spaces, Math. Nachr., 268 (2004) 74 – 95.

[19] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer-Verlag, 1977.

[20] J. Lindenstrauss, On projections with norm 1 - an example, Proc. Amer. Math. Soc. 15 (1964) 403 – 406.

[21] P.D. Morris, E. W. Cheney, On the existence and characterization of minimal projections, Journ. Reine und Angew. Math., 270 (1974) 61 – 76.

[22] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, 1983.

[23] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co., Ltd., Tokyo, 1950.

[24] P. L. Papini and I. Singer, Best co-approximation in normed linear spaces, Monatsh. Math. 88 (1979) 27 – 44.

[25] B. Randrianantoanina, Norm one projections in Banach spaces, Taiwanese J. Math. 5 (2001) 35 – 95.

[26] S.Reich, Product formulas, nonlinear semigroups and accretive operators, Journal of Functional Analysis 36 (1980) 147 - 168.

[27] U. Westphal, Cosuns in l^{p}(n), J. Approx. Theory 54 (1988) 287 – 305.

[28] J. Woo, On modular sequence spaces, Studia Math. 58 (1973), 271–289. Authors’ addresses:

Grzegorz Lewicki

Department of Mathematics

Jagiellonian University

30-059 Kraków, Reymonta 4, Poland E-mail: lewicki@im.uj.edu.pl Giulio Trombetta Department of Mathematics University of Calabria 87036 Arcavacata di Rende, Cosenza, Italy E-mail: trombetta@unical.it