On Convex Total Bounded Sets in the Space of Measurable Functions

Abstract

We estimate the measure of nonconvex total boundedness in terms of simpler quantitative characteristics in the space of measurable functions  𝐿₀. A Fréchet-Smulian type compactness criterion for convexly totally bounded subsets of 𝐿₀ is established.

1. Introduction

In 1988, Idzik [1] proved that the answer to the well-known Schauder's problem [2, Problem 54]: does every continuous map 𝑓∶𝐾→𝐾  defined on a convex compact subset of a Hausdorff topological linear space have a fixed point? is affirmative if 𝐾 is convexly totally bounded.

This notion was introduced by Idzik [1]: a subset 𝐾 of a topological linear space 𝑋 is said to be convexly totally bounded (ctb for short), if for every 0-neighborhood 𝑈 there are 𝑥₁,…,𝑥_𝑛∈𝐾 and convex subsets 𝐶₁,…,𝐶_𝑛 of 𝑈 such that 𝐾⊆⋃^𝑛𝑖=1(𝑥_𝑖+𝐶_𝑖). If 𝑋 is locally convex, every convex compact subset of 𝑋 is ctb. This is not true, in general, if 𝑋 is nonlocally convex (see [3–5]).

In 1993, De Pascale et al. [3] defined the measure of nonconvex total boundedness, modelled on Idzik's concept, that may be regarded as the analogue of the well-known notion of Hausdorff measure of noncompactness in nonlocally convex linear spaces. The above notions of ctb set and of nonconvex total boundedness are especially useful working in the setting of nonlocally convex topological linear spaces (see, e.g., [6–8]).

Let (𝐸,‖⋅‖_𝐸) be a normed space, Ω a nonempty set, and 𝒫(Ω) the power set of Ω. The space 𝐿₀ is an 𝐹-normed linear space of 𝐸-valued functions defined on Ω which depends on an algebra 𝒜 in 𝒫(Ω) and a submeasure 𝜂∶𝒫(Ω)→[0,+∞].

We observe that the space 𝐿₀ defined above is a generalization of the space of measurable functions introduced in [9, Chapter III], in order to develop the integration theory with respect to finitely additive measures. Recall that given 𝑀⊆𝐿₀, the Hausdorff measure of noncompactness 𝛾(𝑀) of 𝑀 is defined by 𝛾(𝑀)∶=inf𝑀⊆𝜀>_𝑛0∶thereexistfunctions𝑓₁,…,𝑓_𝑛∈𝐿₀suchthat_𝑖=1𝑓_𝑖+𝐵_𝜀𝐿₀,(1.1) where 𝐵_𝜀(𝐿₀)∶={𝑓∈𝐿₀∶‖𝑓‖₀≤𝜀}. In [10, 11], to estimate 𝛾(𝑀) are used two quantitative characteristics 𝜆(𝑀) and 𝜔(𝑀) which measure, respectively, the degree of non equi-quasiboundedness and the degree of non equi-measurability of 𝑀.

The main purpose of this note is to estimate the measure of nonconvex total boundedness in 𝐿₀ and to characterize the convexly totally bounded subsets of 𝐿₀. At this end we introduce two quantitative characteristics 𝜆_𝑐(𝑀) and 𝜔_𝑐(𝑀) involving convex sets, which measure the degree of nonconvex equi-quasiboundedness and the degree of nonconvex equi-measurability of 𝑀, respectively. Then we establish some inequalities between 𝛾_𝑐(𝑀), 𝜆_𝑐(𝑀), 𝜔_𝑐(𝑀), and 𝜆(𝑀) that give, as a special case, a Fréchet-Smulian type convex total boundedness criterion in the space 𝐿₀. This generalizes previous results of Trombetta [12]. Finally, we point out that it is not so clear if the Schauder's problem has been solved in its generality. In particular, the proof given by Cauty in [13] contains some unsolved gaps (see [14, 15], Mathscinet review of [16], and Zentralblatt Math review of [17]). However, the results of this paper are meant to be independent from the Schauder's problem.

2. Definitions and Preliminaries

For the remainder of this section we present some definitions and known results which will be needed throughout this paper.

We use the convention inf∅∶=+∞. Moreover, for two sets 𝐴 and 𝐵, we denote by 𝐵^𝐴 the set of all maps from 𝐴 to 𝐵. We recall the definition of the space 𝐿₀ (see [11, 18]). Let (𝐸,‖⋅‖_𝐸) be a normed space, Ω a nonempty set, 𝒜 an algebra in the power set ℙ(Ω) of Ω, and 𝜂∶ℙ(Ω)→[0,∞] a submeasure. Then‖𝑓‖₀∶=inf𝑎>0∶𝜂𝑥∈Ω∶‖𝑓(𝑥)‖_𝐸≥𝑎≤𝑎(2.1) defines a group pseudonorm on 𝐸^Ω, that is, ‖0‖₀=0, ‖−𝑓‖₀=‖𝑓‖₀, ‖𝑓+𝑔‖₀≤‖𝑓‖₀+‖𝑔‖₀, for all 𝑓,𝑔∈𝐸^Ω. Let 𝐿₀∶=𝐿₀(Ω,𝒜,𝐸,𝜂) be the closure of the linear space 𝑆∶=span{𝑦𝜒_𝐴∶𝑦∈𝐸and𝐴∈𝒜} of 𝐸-valued 𝒜-simple functions in (𝐸^Ω,‖⋅‖₀), where 𝜒_𝐴 is the characteristics function of 𝐴. Identification of functions 𝑓,𝑔∈𝐸^Ω for which ‖𝑓−𝑔‖₀=0 turns (𝐿₀,‖⋅‖₀) (the space of measurable functions) into an 𝐹-normed linear space in the sense of [19, page 38], that is, ‖𝑓+𝑔‖₀≤‖𝑓‖₀+‖𝑔‖₀, lim_𝑛→∞‖(1/𝑛)𝑓‖₀=0, ‖𝜆𝑓‖₀≤‖𝑓‖₀ for 𝑓,𝑔∈𝐿₀ and |𝜆|≤1.

Let 𝑀⊆𝐿₀. The measure of nonconvex total boundedness 𝛾_𝑐(𝑀) of 𝑀 is defined by 𝛾_𝑐(𝑀)∶=inf𝐶𝜀>0∶thereexistfunctions𝑓₁,…,𝑓_𝑛∈𝐿₀andconvexsubsets₁,…,𝐶_𝑛of𝐵_𝜀𝐿₀suchthat𝑀⊆^𝑛_𝑖=1𝑓_𝑖+𝐶_𝑖.(2.2) Clearly 𝑀 is ctb if and only if 𝛾_𝑐(𝑀)=0.

Let 𝐴₁,…,𝐴_𝑚∈𝒜 be a partition of Ω. We set 𝑆𝐴₁,…,𝐴_𝑚∶=𝑠∈𝑆∶𝑠=^𝑚_𝑖=1𝑦_𝑖𝜒_𝐴𝑖,where𝑦_𝑖∈𝐸for𝑖=1,…,𝑚.(2.3) In [11] the following two quantitative characteristic 𝜆 and 𝜔 are used to estimate 𝛾 in 𝐿₀: 𝜔𝜆((𝑀𝑀))∶∶==iinnff𝜀𝜀>>00∶∶tthheerreeeexxiissttssaafipanrittietisounb𝐴se₁t𝐺of𝐸suchthat𝑀⊆𝐺^Ω∩𝑆+𝐵_𝜀𝐿₀,suchthat𝑀⊆𝑆𝐴₁,…,𝐴_𝑚∈𝒜ofΩ,…,𝐴_𝑚+𝐵_𝜀𝐿₀.(2.4)

The set 𝑀 is called equi-quasibounded if 𝜆(𝑀)=0 and equi-measurable if 𝜔(𝑀)=0.

Theorem 2.1 (see [11, Theorem 2.2.2]). Let 𝑀⊆𝐿₀. Then max{𝜆(𝑀),𝜔(𝑀)}≤𝛾(𝑀)≤𝜆(𝑀)+2𝜔(𝑀).(2.5) In particular, 𝑀 is totally bounded if and only if 𝜆(𝑀)=𝜔(𝑀)=0.

Moreover, in [18], it is defined the following quantitative characteristic 𝜎(𝑀) which is useful for the calculation of 𝜆(𝑀):𝜎(𝑀)∶=inf{th𝜀e>re0is∶𝐷th_𝑓ereexistsafinitesubset𝐺of𝐸suchthatforall𝑓∈𝑀⊆Ωwith𝜂𝐷_𝑓≤𝜀and𝑓Ω⧵𝐷_𝑓⊆𝐺+𝐵_𝜀(𝐸),(2.6) where 𝐵_𝜀(𝐸)∶={𝑦∈𝐸∶‖𝑦‖_𝐸≤𝜀}.

The following result was established in [18, Proposition 2.1].

Proposition 2.2. Let 𝑀⊆𝐿₀. Then 𝜆(𝑀)=𝜎(𝑀).

We omit the proof of the following proposition which is similar to the proof of Proposition 2.6 of [12].

Proposition 2.3. Let 𝑀⊆𝐿₀. Then 𝛾_𝑐(𝑀)=inf𝐶𝜀>0∶thereexistfunctions𝑠₁,…,𝑠_𝑛∈𝑆andconvexsubsets₁,…,𝐶_𝑛of𝐵_𝜀𝐿₀suchthat𝑀⊆^𝑛_𝑖=1𝑠_𝑖+𝐶_𝑖.(2.7)

3. Inequalities in the Space 𝐿₀

In order to estimate the measure of nonconvex total boundedness in 𝐿₀, we introduce the following two quantitative characteristics.

Definition 3.1. Let 𝑀⊆𝐿₀. We define the following: 𝜆_𝑐(𝑀)∶=inf𝐶𝜀>0∶thereexistafinitesubset𝐺of𝐸andconvexsubsets₁,…,𝐶_𝑛of𝐵_𝜀𝐿₀suchthat𝑀⊆𝐺^Ω∩𝑆+^𝑛_𝑖=1𝐶_𝑖𝜔_𝑐,(𝑀)∶=inf𝐶𝜀>0∶thereexistapartition𝐴₁,…,𝐴_𝑚∈𝒜ofΩandconvexsubsets₁,…,𝐶_𝑛of𝐵_𝜀𝐿₀suchthat𝑀⊆𝑆𝐴₁,…,𝐴_𝑚+^𝑛_𝑖=1𝐶_𝑖.(3.1)

We call 𝑀 convexly equi-quasibounded if 𝜆_𝑐(𝑀)=0 and convexly equi-measurable if 𝜔_𝑐(𝑀)=0.

We observe that if 𝐸=ℝ, then the quantitative characteristics 𝜆_𝑐 and 𝜔_𝑐 coincide with those introduced in [12].

We point out that the request of convexity plays a crucial role in the definition of the parameters 𝜆_𝑐 and 𝜔_𝑐, whereas it was not involved in the definition of 𝜆 and 𝜔. We illustrate this with the following example.

Example 3.2. Let 𝐿₀∶=𝐿₀([0,+∞[,𝒜,𝐸,𝜂), where 𝒜 is the algebra of all Lebesgue-measurable subsets of the interval [0,+∞[ and 𝜂|_𝒜 the Lebesgue measure. Let 𝐼₁∶=[0,1[, 𝐼_𝑛∶=[∑^{𝑛−1𝑘=1}(1/𝑘),∑^𝑛𝑘=1(1/𝑘)[ for 𝑛≥2, and 𝑀_𝑛∶={𝑦𝜒_𝐼𝑛∶𝑦∈𝐸} for 𝑛≥1. If 𝑀∶=⋃^∞𝑛=1𝑀_𝑛, then 𝜆(𝑀)=1, 𝜔(𝑀)=0, 𝜆_𝑐(𝑀)=𝜔_𝑐(𝑀)=+∞.

Proof. It is easy to check that 𝜆(𝑀)=1 and 𝜔(𝑀)=0. We are going to prove that 𝜆_𝑐(𝑀)=+∞. On the contrary, suppose that 𝜆_𝑐(𝑀)<𝛼<+∞ then there exist a finite set 𝐺={𝑧₁,…,𝑧_𝑝}⊆𝐸 and convex sets 𝐶₁,…,𝐶_𝑚⊆𝐵_𝛼(𝐿₀) such that 𝑀⊆𝐺^Ω∩𝑆+^𝑚_𝑗=1𝐶_𝑗.(3.2) Set 𝑐∶=max{‖𝑧_𝑙‖_𝐸∶𝑙=1,…,𝑝}. We have ‖𝑠‖₀≤𝑐 for all 𝑠∈𝐺^Ω∩𝑆. Fix 𝛼>𝑐+𝛼, a natural number 𝑛 such that ∑^𝑛𝑛=1(1/𝑛)>𝑚𝛼, and 𝑦∈𝐸 with ‖𝑦‖_𝐸>𝑛⋅𝛼.
Put 𝑆_𝑗𝑦𝑆_𝑦∶=𝑓=𝑦𝜒_𝐼𝑛∶𝑛=1,2,…,𝑛,∶=𝑓∈𝑆_𝑦∶thereis𝑠_𝑓∈𝐺^Ω∩𝑆suchthat𝑓−𝑠_𝑓∈𝐶_𝑗(3.3) for 𝑗=1,…,𝑚.
Then it is easy to see that there exist 𝑗∈{1,…,𝑚} and a subfamily {𝐼_𝑛1,…,𝐼_𝑛𝑘} of {𝐼₁,…,𝐼_𝑛} such that 𝑓_𝑛𝑘∈𝑆^𝑗_𝑦 for 𝑘=1,…,𝑘 and 𝜇(⋃^𝑘𝑘=1𝐼_𝑛𝑘)≥𝛼. Moreover, a straightforward computation shows that ‖‖‖‖_𝑘_𝑘=11𝑘(𝑓^𝑛𝑘−𝑠_𝑓𝑛𝑘)‖‖‖‖₀>𝛼,(3.4) which contradicts the convexity of the set 𝐶_𝑗. Since 𝜆_𝑐(𝑀)=+∞, the equality 𝜔_𝑐(𝑀)=+∞ is a consequence of Theorem 3.4.

The following lemma is crucial in the proof of Theorem 3.4.

Lemma 3.3. Let 𝐴₁,…,𝐴_𝑛∈𝒜 be a partition of Ω and 𝐻⊆𝑆(𝐴₁,…,𝐴_𝑛). Then 𝜆(𝐻)=𝛾_𝑐(𝐻).

Proof. Obviously, 𝛾(𝐻)≤𝛾_𝑐(𝐻). Since 𝜔(𝐻)=0, it follows from Theorem 2.1 and Proposition 2.2 that 𝜆(𝐻)=𝛾(𝐻)=𝜎(𝐻). Then it is sufficient to prove the inequality 𝛾_𝑐(𝐻)≤𝜎(𝐻) which is trivial if 𝜎(𝐻)=𝜂(Ω).
Assume that 𝜎(𝐻)<𝛼<𝜂(Ω). By the definition of 𝜎 we can find a finite set 𝐺⊆𝐸, containing the origin 0∈𝐸 and sets 𝐷₀∶=∅, 𝐷₁,…,𝐷_𝑟⊆Ω, with 𝜂(𝐷_𝑗)≤𝛼 for 𝑗=1,…,𝑟, such that(i)each 𝐷_𝑗 for 𝑗=1,…,𝑟 is the union of the members of a proper subfamily depending on 𝑗 of the partition {𝐴₁,…,𝐴_𝑛};(ii)for each 𝑠∈𝐻, there is 𝑗∈{0,…,𝑟} such that 𝑠(Ω⧵𝐷_𝑗)⊆𝐺+𝐵_𝛼(𝐸).
For 𝑗=0,…,𝑟 we consider the following convex subsets of 𝐵_𝛼(𝐿₀): 𝐶^𝑗𝛼∶=𝑠∈𝑆𝐴₁,…,𝐴_𝑛∶𝑠Ω⧵𝐷_𝑗⊆𝐵_𝛼(𝐸),(3.5) and we set 𝐻_𝑗∶=𝑠∈𝐻∶𝑠Ω⧵𝐷_𝑗⊆𝐺+𝐵_𝛼(𝐸).(3.6) We will prove that 𝐻=^𝑟_𝑗=0𝐻_𝑗⊆𝐺^Ω∩𝑆𝐴₁,…,𝐴_𝑛+^𝑟_𝑗=0𝐶^𝑗𝛼.(3.7)
It follows that 𝛾_𝑐(𝐻)≤𝛼, and therefore 𝛾_𝑐(𝐻)≤𝜎(𝐻).
Let 𝑠=∑^𝑛𝑖=1𝑦_𝑖𝜒_𝐴𝑖∈𝐻. Suppose 𝑠∈𝐻_𝑗, 𝑗≥1 and 𝐷_𝑗=⋃^𝑘𝑙=1𝐴_𝑖𝑙, where {𝐴_𝑖1,…,𝐴_𝑖𝑘} is a proper subfamily of the partition {𝐴₁,…,𝐴_𝑛}.
Put {𝐴_𝑖𝑘+1,…,𝐴_𝑖𝑛}∶={𝐴₁,…,𝐴_𝑛}⧵{𝐴_𝑖1,…,𝐴_𝑖𝑘}.
We have 𝑠Ω⧵𝐷_𝑗=𝑦_𝑖𝑘+1,…,𝑦_𝑖𝑛⊆𝐺+𝐵^𝛼(𝐸).(3.8) Hence, for all 𝑙∈{𝑘+1,…,𝑛}, there is 𝑧_𝑖𝑙∈𝐺 such that 𝑦_𝑖𝑙−𝑧_𝑖𝑙∈𝐵_𝛼(𝐸). Then 𝑠=𝜑+ℎ, where 𝜑∶=^𝑛_𝑙=_𝑘+1ℎ∶=_𝑘𝑧_𝑖𝑙𝜒_𝐴𝑖𝑙∈𝐺^Ω∩𝑆𝐴₁,…,𝐴_𝑛,_𝑙=1𝑦_𝑖𝑙𝜒_𝐴𝑖𝑙+^𝑛_𝑙=_𝑘+1𝑦_𝑖l−𝑧_𝑖𝑙𝜒_𝐴𝑖𝑙∈𝐶^𝑗𝛼.(3.9) Therefore, 𝑠∈𝐺^Ω∩𝑆𝐴₁,…,𝐴_𝑛+𝐶^𝑗𝛼.(3.10) Similarly, if 𝑠∈𝐻₀, we can prove that 𝑠∈𝐺^Ω∩𝑆𝐴₁,…,𝐴_𝑛+𝐶^0𝛼.(3.11) Thus, (3.7) immediately follows from (3.10) and (3.11).

We are now in a position to prove the main result of this note.

Theorem 3.4. Let 𝑀⊆𝐿₀. Then, max𝜆_𝑐(𝑀),𝜔_𝑐(𝑀)≤𝛾_𝑐(𝑀)≤𝜆(𝑀)+2𝜔_𝑐(𝑀).(3.12)

Proof. We first prove the left inequality which is trivial if 𝛾_𝑐(𝑀)=+∞. Assume that 𝛾_𝑐(𝑀)<𝛼<+∞. By Proposition 2.3, there are functions 𝑠₁,…,𝑠_𝑛∈𝑆 and convex sets 𝐶₁,…,𝐶_𝑛 in 𝐵_𝛼(𝐿₀) such that 𝑀⊆^𝑛_𝑖=1𝑠_𝑖+𝐶_𝑖.(3.13)
Put 𝐹∶=⋃^𝑛𝑖=1𝑠_𝑖(Ω) and let 𝐴₁,…,𝐴_𝑚∈𝒜 be a partition of Ω such that 𝑠_𝑖|_𝐴𝑗 is constant for 𝑖=1,…,𝑛 and 𝑗=𝑖,…,𝑚. Then, 𝑀⊆𝐹^Ω∩𝑆𝐴₁,…,𝐴_𝑚+^𝑛_𝑖=1𝐶_𝑖,(3.14) hence 𝜆_𝑐(𝑀)<𝛼, and 𝜔_𝑐(𝑀)<𝛼.
Therefore, max𝜆_𝑐(𝑀),𝜔_𝑐(𝑀)≤𝛾_𝑐(𝑀).(∗) We now prove the right inequality. Clearly, it is true if 𝜆(𝑀)=+∞ or 𝜔_𝑐(𝑀)=+∞. Assume that 𝜔_𝑐(𝑀)<𝛽<+∞. By the definition of 𝜔_𝑐, we can find a partition 𝐴₁,…,𝐴_𝑚∈𝒜 and convex sets 𝐾₁,…,𝐾_𝑚 of 𝐵_𝛽(𝐿₀) such that 𝑀⊆𝑆𝐴₁,…,𝐴_𝑚+^𝑚_𝑗=1𝐾_𝑗.(3.15) Set 𝐻∶=𝑀−^𝑚_𝑗=1𝐾_𝑗∩𝑆𝐴¹,…,𝐴_𝑚,(3.16) then we have 𝜆(𝐻)≤𝜆(𝑀)+𝛽. It easy to see that 𝑀⊆𝐻+^𝑚_𝑗=1𝐾_𝑗.(3.17)
Therefore, by Lemma 3.3, we have that 𝛾_𝑐(𝑀)≤𝛾_𝑐(𝐻)+𝛽=𝜆(𝐻)+𝛽≤𝜆(𝑀)+𝛽+𝛽,(3.18) and so 𝛾_𝑐(𝑀)≤𝜆(𝑀)+2𝜔_𝑐(𝑀).(3.19) The proof is complete.

As a corollary of Theorem 3.4, we obtain the following Fréchet-Smulian type convex total boundedness criterion.

Corollary 3.5. A subset 𝑀 of 𝐿₀ is ctb if and only if 𝜆(𝑀)=𝜔_𝑐(𝑀)=0.

Remark 3.6. In [12], Trombetta proved that max𝜆_𝑐(𝑀),𝜔_𝑐(𝑀)≤𝛾_𝑐(𝑀)≤𝜆_𝑐(𝑀)+2𝜔_𝑐(𝑀)(3.20) for a subset 𝑀⊆𝐿₀(Ω,𝒜,ℝ,𝜂). Since 𝜆(𝑀)≤𝜆_𝑐(𝑀), Theorem 3.4 improves and generalizes to 𝐸-valued case the above inequalities.

We point out that the approach used in the scalar case [12] in order to prove Theorem 3.7 cannot be used in our framework. The crucial difference is in the proof of Lemma 3.3. In fact, if dim(𝐸)=+∞, it might exist some 𝑖∈{1,…,𝑛} such that the set 𝐻(𝐴_𝑖)∶={𝑠(𝐴_𝑖)∶𝑠∈𝐻} is bounded but not necessarily totally bounded.

The following example shows how the value of the parameters 𝜆 and 𝜆_𝑐 changes when passing from the scalar case to the 𝐸-valued case.

Example 3.7. Let 𝑀∶={𝑦𝜒_Ω∶𝑦∈𝐵(𝐸)}, where 𝐵(𝐸)={𝑦∈𝐸∶‖𝑦‖_𝐸≤1}. Then 𝜆(𝑀)=𝜆_𝑐(𝑀)=0 if 𝐸 is finite dimensional, 𝜆(𝑀)=𝜆_𝑐(𝑀)=1, otherwise. Moreover, since 𝜔_𝑐(𝑀)=0, we have 𝜆(𝑀)=𝜆_𝑐(𝑀)=𝛾_𝑐(𝑀).

Corollary 3.8. Let 𝑀⊆𝐿₀. Then, max𝜆(𝑀),𝜔_𝑐(𝑀)≤≤mmaaxx𝜆𝜆_𝑐𝑀,𝜔_𝑐𝑀≤𝜆(𝑀)+2𝜔𝑀_𝑐,𝜔_𝑐𝑀(𝑀).(3.21)

Proof. It is sufficient to observe that 𝛾_𝑐(𝑀)=𝛾_𝑐(𝑀).
In particular, 𝑀 ctb implies 𝜆(𝑀)=𝜆(𝑀)=0, 𝜆_𝑐(𝑀)=𝜆_𝑐(𝑀)=0, and 𝜔_𝑐(𝑀)=𝜔_𝑐(𝑀)=0.

The next two corollaries of Theorem 3.4 are useful in order to compute or to estimate 𝛾_𝑐 in particular classes of subsets of 𝐿₀. Moreover, the second one generalizes [12, Proposition 3.10].

Corollary 3.9. Let 𝑀 be a convexly equi-measurable subset of 𝐿₀. Then, 𝜆(𝑀)=𝛾_𝑐(𝑀).

Corollary 3.10. Let 𝑀 be an equi-quasibounded subset of 𝐿₀. Then, max𝜆_𝑐(𝑀),𝜔_𝑐(𝑀)≤𝛾_𝑐(𝑀)≤2𝜔_𝑐(𝑀).(∗∗)

We observe that if 𝐾 is a totally bounded subset of 𝐸 and 𝑀∶={𝑓∈𝐿₀∶𝑓(Ω)⊆𝐾}, then, since 𝜆(𝑀)=0, the inequalities (3.21) are true for 𝑀.

Example 3.11. Let 𝐿₀ be the space of Example 3.2 and assume that 𝐸 is the Banach space 𝑙^∞ of all sequences 𝑦=(𝜉₁,𝜉₂,…) with finite norm ‖𝑦‖_∞∶=sup{|𝜉_𝑛|∶𝑛=1,2,…}. If 𝐾∶={𝑦∈𝑙_∞∶|𝜉_𝑛|≤1/𝑛for𝑛=1,2,…} and 𝑀∶={𝑓∈𝐿₀∶𝑓(Ω)⊆𝐾}, it is easy to prove that 𝜆_𝑐(𝑀)=0 and 𝜔_𝑐(𝑀)=𝛾_𝑐(𝑀)=1. If 𝐵(𝑙^∞) is the closed unit ball of 𝑙^∞ and 𝑀∶={𝑓∈𝐿₀∶𝑓(Ω)⊆𝐵(𝑙^∞)}, the set 𝑀 satisfies 𝜆(𝑀)=𝜆_𝑐(𝑀)=𝛾_𝑐(𝑀)=𝜔_𝑐(𝑀)=1.