The Uniform Boundedness Theorem in Asymmetric Normed Spaces

Abstract

We obtain a uniform boundedness type theorem in the frame of asymmetric normed spaces. The classical result for normed spaces follows as a particular case.

1. Introduction

Throughout this paper the letters ℝ and ℝ⁺ will denote the set of real numbers and the set of nonnegative real numbers, respectively.

The book of Aliprantis and Border [1] provides a good basic reference for functional analysis in our context.

Let 𝑋 be a real vector space. A function 𝑝∶𝑋→ℝ⁺ is said to be an asymmetric norm on 𝑋 [2, 3] if for all 𝑥,𝑦∈𝑋, and 𝑟∈ℝ⁺,(i)𝑝(𝑥)=𝑝(−𝑥)=0 if and only if 𝑥=0;(ii)𝑝(𝑟𝑥)=𝑟𝑝(𝑥);(iii)𝑝(𝑥+𝑦)≤𝑝(𝑥)+𝑝(𝑦).

The pair (𝑋,𝑝) is called an asymmetric normed space.

Asymmetric norms are also called quasinorms in [4–6], and nonsymmetric norms in [7].

If 𝑝 is an asymmetric norm on 𝑋, then the function 𝑝⁻¹ defined on 𝑋 by 𝑝⁻¹(𝑥)=𝑝(−𝑥) is also an asymmetric norm on 𝑋, called the conjugate of 𝑝. The function 𝑝^𝑠 defined on 𝑋 by 𝑝^𝑠(𝑥)=max{𝑝(𝑥),𝑝⁻¹(𝑥)} is a norm on 𝑋. We say that (𝑋,𝑝) is a bi-Banach space if (𝑋,𝑝^𝑠) is a Banach space [3]. The following is a simple instance of a biBanach space.

Example 1.1. Denote by 𝑢 the function defined on ℝ by 𝑢(𝑥)=𝑥∨0 for all 𝑥∈ℝ. Then 𝑢 is an asymmetric norm on ℝ such that 𝑢^𝑠 is the Euclidean norm on ℝ, that is, (ℝ,𝑢^𝑠) is the Euclidean normed space (ℝ,|⋅|). Hence (ℝ,𝑢) is a biBanach space.

By a quasimetric on a nonempty set 𝑋 we mean a function 𝑑∶𝑋×𝑋→ℝ⁺ such that for all 𝑥,𝑦,𝑧∈𝑋∶ (i) 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥)=0⇔𝑥=𝑦, and (ii) 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦).

The function 𝑑⁻¹ defined on 𝑋×𝑋 by 𝑑⁻¹(𝑥,𝑦)=𝑑(𝑦,𝑥) is a quasi-metric on 𝑋 called the conjugate of 𝑑.

Each quasi-metric 𝑑 on 𝑋 induces a 𝑇₀ topology 𝑇(𝑑) on 𝑋 which has as a base the family of the balls {𝐵_𝑑(𝑥,𝑟)∶𝑥∈𝑋,𝑟>0}, where 𝐵_𝑑(𝑥,𝑟)={𝑦∈𝑋∶𝑑(𝑥,𝑦)<𝑟}.

Each asymmetric norm 𝑝 on a real vector space 𝑋 induces a quasi-metric 𝑑_𝑝 defined by 𝑑_𝑝(𝑥,𝑦)=𝑝(𝑦−𝑥), for all 𝑥,𝑦∈𝑋. We refer to the topology 𝑇(𝑑_𝑝) as the topology induced by 𝑝. The terms 𝑝-neighborhood, 𝑝-open, 𝑝-closed, and so forth will refer to the corresponding topological concepts with respect to that topology. The ball 𝐵_𝑑𝑝(𝑥,𝑟) will be simply denoted by 𝐵_𝑝(𝑥,𝑟).

It was shown in [5] that for any normed lattice (𝑋,‖⋅‖), the function 𝑝 defined on 𝑋 by 𝑝(𝑥)=‖𝑥⁺‖, with 𝑥⁺=𝑥∨0, is an asymmetric norm on 𝑋 and the norm 𝑝^𝑠 is equivalent to the norm ‖⋅‖. Moreover, 𝑝 determines both the topology and order of 𝑋. We will refer to 𝑝 as the asymmetric norm associated to(𝑋,‖⋅‖).

It seems interesting to point out that the recent development of the theory of asymmetric normed spaces has been motivated, in great part, by their applications. In fact, asymmetric norms (in particular, those that are associated to normed lattices) and other related nonsymmetric structures from topological algebra and functional analysis have been applied to construct suitable mathematical models in theoretical computer science [2, 8–10] as well as to some questions in approximation theory [6, 7, 11, 12].

The asymmetric normed spaces share some properties with usual normed spaces but there are also significant differences between them. In the last decade, several papers on general topology and functional analysis have been published in order to extend well-known results of the theory of normed spaces to the framework of asymmetric normed spaces (see, e.g., [3, 4, 11, 13–17]).

In this sense, the recent book of Cobzas [18] collects in a unified way the most interesting results obtained up to date in the context of nonsymmetric topology and fuctional analysis. Furthermore, in this monograph, the author also presents new results that significantly enrich the theory of asymmetric normed spaces. One of these new results which appears in the book is the uniform boundedness theorem that extends the classical one for normed spaces. In our terminology, this result is formulated as follows.

Let (𝑋,𝑝) be a right K-complete asymmetric normed space, and let (𝑌,𝑞) be an asymmetric normed space. If ℱ is a family of continuous linear operators from (𝑋,𝑝) to (𝑌,𝑞) such that for each 𝑥∈𝑋, there exist 𝑏_𝑥>0 and 𝑐_𝑥>0 with 𝑞(𝑓(𝑥))≤𝑏_𝑥 and 𝑞⁻¹(𝑓(𝑥))≤𝑐_𝑥 for all 𝑓∈ℱ, then there exist 𝑏>0 and 𝑐>0 such that sup𝑞𝑓(𝑥)∶𝑝⁻¹(𝑥)≤1≤𝑏,sup𝑞⁻¹𝑓(𝑥)∶𝑝(𝑥)≤1≤𝑐, (1.1) for all 𝑓∈ℱ.

The condition of right K-completeness for (𝑋,𝑝) leaves outside the scope of this theorem an important class of asymmetric normed spaces, the asymmetric normed spaces associated to normed lattices because these spaces are right K-complete only for the trivial case [13]. In this paper, we give a uniform boundedness type theorem in the setting of asymmetric normed spaces which extends the classical result for normed spaces and it is less restrictive than Cobzas' theorem. For this purpose we introduce the notion of quasi-metric space of the half second category. These spaces play in our context a similar role that the metric spaces of the second category have played in the realm of normed spaces.

2. The Results

The result for normed linear spaces usually called Uniform Boundedness or Banach-Steinhauss theorem is formulated as follows.

Theorem 2.1 (see [1]). Let 𝑋 be a Banach space and let 𝑌 be a normed space. If ℱ is a family of continuous linear operators from 𝑋 to 𝑌 such that for each 𝑥∈𝑋 there exists 𝑏_𝑥>0 with ‖𝑓(𝑥)‖≤𝑏_𝑥 for all 𝑓∈ℱ, then there exists 𝑏>0 such that ‖𝑓‖≤𝑏 for all 𝑓∈ℱ.

A natural way of extending the preceding result to the context of asymmetric normed spaces consists of replacing Banach space by biBanach asymmetric normed spaces. Thus one may conjecture that the next result would be desirable.

Let 𝑋 be a biBanach asymmetric normed space and let 𝑌 be an asymmetric normed space. If ℱ is a family of continuous linear operators from (𝑋,𝑝) to (𝑌,𝑞) such that for each 𝑥∈𝑋 there exists 𝑏_𝑥>0 with 𝑞(𝑓(𝑥))≤𝑏_𝑥 for all 𝑓∈ℱ, then there exists 𝑏>0 such that sup𝑞𝑓(𝑥)∶𝑝(𝑥)≤1≤𝑏, (2.1) for all 𝑓∈ℱ.

Nevertheless, the following example shows that such a result does not hold in our context.

Example 2.2. Let 𝑋_𝑘={(𝑥_𝑛)∈𝑙₁∶2𝑘𝑥_2𝑘−1+𝑥_2𝑘=0}. Since 𝑋_𝑘 is a closed linear subspace of 𝑙₁ for every 𝑘∈ℕ, the subspace 𝑋=^∞_𝑘=1𝑋_𝑘, (2.2) is a closed linear subspace of 𝑙₁. We consider the asymmetric normed space (𝑙₁,𝑝), where 𝑝(𝑥)=𝑥⁺₁=^∞_𝑛=1𝑥⁺_𝑛. (2.3)
Since the norm 𝑝^𝑠 is equivalent to the norm ‖⋅‖₁ on 𝑙₁, we have that 𝑋 is a closed linear subspace of the Banach space (𝑙₁,𝑝^𝑠) and then (𝑋,𝑝^𝑠) is a Banach space. Therefore (𝑋,𝑝) is a biBanach space.
Let 𝑓_𝑛∶𝑋→ℝ be given by 𝑓_𝑛(𝑥)=(2𝑛+1)𝑥_2𝑛−1 for every 𝑛∈ℕ. Since 𝑓_𝑛(𝑥)=(2𝑛+1)𝑥_2𝑛−1≤(2𝑛+1)𝑥⁺_2𝑛−1≤(2𝑛+1)𝑝(𝑥), (2.4) we have that 𝑓_𝑛 is a continuous linear map from (𝑋,𝑝) to (ℝ,𝑢), for every 𝑛∈ℕ.
If 𝑥∈𝑋, then ‖𝑥‖₁=∑^∞_𝑛=1(2𝑛+1)|𝑥_2𝑛−1|, so that (2𝑛+1)𝑥_2𝑛−1≤(2𝑛+1)𝑥_2𝑛−1≤‖𝑥‖₁. (2.5) Therefore, 𝑓_𝑛(𝑥)=(2𝑛+1)𝑥_2𝑛−1≤‖𝑥‖₁, for every 𝑛∈ℕ.
Now, we will prove that sup{𝑓_𝑛(𝑥)∶𝑝(𝑥)≤1}=2𝑛+1, for every 𝑛∈ℕ. Indeed, if 𝑝(𝑥)≤1, then 𝑓_𝑛(𝑥)≤2𝑛+1. If we consider 𝑥=(𝑥^𝑛_𝑖)^∞_𝑖=1 such that 𝑥^𝑛_2𝑛−1=1, 𝑥^𝑛_2𝑛=−2𝑛 and 𝑥^𝑛_𝑖=0 if 𝑖∈ℕ−{2𝑛−1,2𝑛}, then 𝑥∈𝑋,𝑝(𝑥)=1 and 𝑓_𝑛(𝑥)=2𝑛+1. Hence, sup𝑓_𝑛(𝑥)∶𝑝(𝑥)≤1=2𝑛+1. (2.6)
Consequently, supsup𝑓_𝑛(𝑥)∶𝑝(𝑥)≤1∶𝑛∈ℕ=∞. (2.7)

Before introducing the notion of a quasi-metric space of the half second category we recall some pertinent concepts in order to help the reader.

A topological space 𝑋 is said to be of the first category if it is the union of a countable collection of nowhere dense subsets. 𝑋 is said to be of the second category if it is not of the first category (see, e.g., [19, page 348]) From the characterization of nowhere dense subsets given in Proposition 11.13 [19], it follows that a topological space 𝑋 is of the second category if and only if condition 𝑋=⋃^∞_𝑛=1𝐸_𝑛 implies int(cl𝐸_𝑚)≠∅ for some 𝑚∈ℕ.

If (𝑋,𝑑) is a quasi-metric space and 𝐴 is a subset of 𝑋, we denote by cl_𝑑⁻¹𝐴 the closure of 𝐴 in the topological space (𝑋,𝑇(𝑑⁻¹)) and by int_𝑑𝐴 the interior of 𝐴 in the topological space (𝑋,𝑇(𝑑)). If (𝑋,𝑝) is an asymmetric normed space and 𝐴 is a subset of 𝑋, we will write cl_𝑝−1𝐴 and int_𝑝𝐴 instead of cl_(𝑑𝑝)⁻¹𝐴 and int_𝑑𝑝𝐴, respectively.

Definition 2.3. We say that a quasi-metric space (𝑋,𝑑) is of the half second category if 𝑋=⋃^∞_𝑛=1𝐸_𝑛, implies int_𝑑(cl_𝑑⁻¹𝐸_𝑚)≠∅ for some 𝑚∈ℕ.

Note that if 𝑑 is a metric on 𝑋, the notion of space of the half second category coincides with the classical notion of space of the second category.

The quasi-metric space (ℝ,𝑑_𝑢) given in Example 1.1 is of the half second category. Indeed, if ℝ=⋃^∞_𝑛=1𝐸_𝑛, take 𝑚∈ℕ with 𝐸_𝑚≠∅. If 𝐸_𝑚=ℝ, then cl_(𝑑𝑢)⁻¹(int_𝑑𝑢𝐸_𝑚)=ℝ. Otherwise, since nonempty proper (𝑑_𝑢)⁻¹-closed subsets of ℝ are of the form ]−∞,𝑎],𝑎∈ℝ, there exists 𝑎_𝑚∈ℝ such that cl_(𝑑𝑢)⁻¹𝐸_𝑚=]−∞,𝑎_𝑚]. Therefore, int_𝑑𝑢(cl_(𝑑𝑢)⁻¹𝐸_𝑚)=]−∞,𝑎_𝑚[.

Note that (ℝ,𝑑_𝑢) is not of the second category. Indeed, in Proposition 1 of [20] it is proved that if (𝑋,‖⋅‖) is a normed lattice and 𝑝(𝑥)=‖𝑥⁺‖, then (𝑋,𝑑_𝑝) is not of the second category.

Lemma 2.4 (uniform boundedness principle). If (𝑋,𝑑) is a quasi-metric space of the half second category and ℱ is a family of real valued lower semicontinuous functions on (𝑋,𝑑⁻¹) such that for each 𝑥∈𝑋 there exists 𝑏_𝑥>0 such that 𝑓(𝑥)≤𝑏_𝑥 for all 𝑓∈ℱ, then there exist a nonempty open set 𝑈 in (𝑋,𝑑) and 𝑏>0 such that 𝑓(𝑥)≤𝑏 for all 𝑓∈ℱ and 𝑥∈𝑈.

Proof. For each 𝑛∈ℕ let 𝐸_𝑛=_𝑓∈ℱ𝑓⁻¹]−∞,𝑛]. (2.8)
Then each 𝐸_𝑛 is closed in (𝑋,𝑑⁻¹). Moreover 𝑋=⋃^∞_𝑛=1𝐸_𝑛. Indeed, by our hypothesis, given 𝑥∈𝑋 there exists 𝑛_𝑥∈ℕ such that 𝑓(𝑥)≤𝑛_𝑥 for all 𝑓∈ℱ, so 𝑥∈𝐸_𝑛𝑥.
Hence, there exists 𝑚∈ℕ such that 𝑈≠∅, where 𝑈=int_𝑑(cl_𝑑⁻¹𝐸_𝑚)=int_𝑑𝐸_𝑚. Then, for each 𝑥∈𝑈 and 𝑓∈ℱ we obtain 𝑓(𝑥)≤𝑚. This completes the proof.

Definition 2.5. We say that an asymmetric normed space (𝑋,𝑝) is of the half second category if the quasi-metric space (𝑋,𝑑_𝑝) is of the half second category.

Theorem 2.6 (uniform boundedness theorem). Let (𝑋,𝑝) and (𝑌,𝑞) be two asymmetric normed spaces such that (𝑋,𝑝) is of the half second category. If ℱ is a family of continuous linear operators from (𝑋,𝑝) to (𝑌,𝑞) such that for each 𝑥∈𝑋 there exists 𝑏_𝑥>0 with 𝑞(𝑓(𝑥))≤𝑏_𝑥 for all 𝑓∈ℱ, then there exists 𝑏>0 such that sup𝑞𝑓(𝑥)∶𝑝(𝑥)≤1≤𝑏, (2.9) for all 𝑓∈ℱ.

Proof. For each 𝑔∈ℱ define a function ℎ_𝑔 by ℎ_𝑔(𝑥)=𝑞(𝑔(𝑥)) for all 𝑥∈𝑋. We first show that ℎ_𝑔 is lower semicontinuous on (𝑋,(𝑑_𝑝)⁻¹). Indeed, let 𝑥∈𝑋 and {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋 such that (𝑑_𝑝)⁻¹(𝑥,𝑥_𝑛)→0. Then 𝑝(𝑥−𝑥_𝑛)→0. By the linear continuity of 𝑔 we deduce that 𝑞(𝑔(𝑥)−𝑔(𝑥_𝑛))→0. From the fact that 𝑞(𝑔(𝑥))−𝑞(𝑔(𝑥_𝑛))≤𝑞(𝑔(𝑥)−𝑔(𝑥_𝑛)) for all 𝑛∈ℕ, it follows that for each 𝜀>0 there is 𝑛₀∈ℕ such that ℎ_𝑔(𝑥)−ℎ_𝑔(𝑥_𝑛)<𝜀. Since 𝑥 is arbitrary, we conclude that ℎ_𝑔 is lower semicontinuous on (𝑋,(𝑑_𝑝)⁻¹).
Put ℋ={ℎ_𝑔∶𝑔∈ℱ}. Since for each 𝑥∈𝑋,ℎ_𝑔(𝑥)=𝑞(𝑔(𝑥))≤𝑏_𝑥 for all 𝑔∈ℱ, we can apply Lemma 2.4 and thus there exists a nonempty 𝑝-open subset 𝑈 of 𝑋 and a 𝛿>0 such that ℎ_𝑔(𝑥)≤𝛿 for all 𝑔∈ℱ and 𝑥∈𝑈.
Fix 𝑧∈𝑈. Then, there exists 𝑟>0 such that 𝐵_𝑝(𝑧,𝑟)⊂𝑈. Take an 𝜀∈]0,𝑟[. Then, 𝐵_𝑝(𝑧,𝜀)⊂𝑈, where 𝐵_𝑝(𝑧,𝜀)={𝑦∈𝑋∶𝑝(𝑦−𝑧)≤𝜀}. Put 𝑏=(𝛿+𝑏_−𝑧)/𝜀, and let 𝑥∈𝑋 such that 𝑝(𝑥)≤1. We will prove that 𝑞(𝑓(𝑥))≤𝑏 for all 𝑓∈ℱ. Indeed, first note that 𝑝(𝜀𝑥+𝑧−𝑧)=𝜀𝑝(𝑥)≤𝜀, so 𝜀𝑥+𝑧∈𝑈.
Now take 𝑓∈ℱ. Then 𝑞𝑓(𝑥=1)𝜀𝑞𝑓(=1𝜀𝑥)𝜀𝑞𝑓(=1𝜀𝑥+𝑧−𝑧)𝜀𝑞𝑓()(≤1𝜀𝑥+𝑧+𝑓−𝑧)𝜀𝑞𝑓()𝑓()=1𝜀𝑥+𝑧+𝑞−𝑧𝜀ℎ_𝑓(𝜀𝑥+𝑧)+ℎ_𝑓(≤1−𝑧)𝜀𝛿+𝑏_−𝑧=𝑏. (2.10)

Since the Banach spaces are of the second category, the classical result for normed space is a corollary of this theorem.

Corollary 2.7. Let 𝑋 be a Banach space and let 𝑌 be a normed space. If ℱ is a family of continuous linear operators from 𝑋 to 𝑌 such that for each 𝑥∈𝑋 there exists 𝑏_𝑥>0 with ‖𝑓(𝑥)‖≤𝑏_𝑥 for all 𝑓∈ℱ, then there exists 𝑏>0 such that ‖𝑓‖≤𝑏 for all 𝑓∈ℱ.

In the remainder of this section we give examples of asymmetric normed spaces of the half second category.

Definition 2.8. Let (𝑋,𝑝) be an asymmetric normed space. (𝑋,𝑝) is right bounded if there is 𝑟>0 such that 𝑟𝐵_𝑝(0,1)⊂𝐵_𝑝𝑠(0,1)+𝜃(0), being 𝜃(0)={𝑦∈𝑋∶𝑝(𝑦)=0}.

This definition is equivalent to Definition 16 of [16].

Remark 2.9. The class of right bounded asymmetric normed spaces contains the asymmetric normed spaces given by normed lattices. Indeed, in Lemma 1 of [14] it is proved that if (𝑋,‖⋅‖) is a normed lattice and 𝑝(𝑥)=‖𝑥⁺‖, then the asymmetric normed space (𝑋,𝑝) is right-bounded with constant 𝑟=1.

Lemma 2.10. Let (𝑋,𝑝) be an asymmetric normed space and let 𝜃(0)={𝑦∈𝑋∶𝑝(𝑦)=0}.(1)If 𝐺 is a 𝑝-open subset of 𝑋, then 𝐺+𝜃(0)⊂𝐺.(2)If 𝐺 is 𝑝⁻¹-open subset of 𝑋, then 𝐺−𝜃(0)⊂𝐺.(3)If 𝐹 is a 𝑝⁻¹-closed subset of 𝑋, then 𝐹+𝜃(0)⊂𝐹(4)If (𝑋,𝑝) right bounded, then for all 𝑎∈𝑋 and for all 𝛿>0, there is 𝑘>0 such that 𝐵_𝑝(𝑎,𝑘)⊂𝐵_𝑝𝑠(𝑎,𝛿)+𝜃(0).

Proof. (1) Let 𝑦∈𝐺+𝜃(0). There exists 𝑥∈𝐺 such that 𝑦−𝑥∈𝜃(0). Since 𝐺 is 𝑝-open, there is 𝜀>0 such that 𝐵_𝑝(𝑥,𝜀)⊂𝐺. Since 𝑝(𝑦−𝑥)=0, we have that (𝑦−𝑥)∈𝐵_𝑝(0,𝜀), then 𝑦∈𝑥+𝐵_𝑝(0,𝜀)=𝐵_𝑝(𝑥,𝜀)⊂𝐺.
(2) Let 𝑦∈𝐺−𝜃(0). There exists 𝑥∈𝐺 such that 𝑦−𝑥∈−𝜃(0), that is, 𝑥−𝑦∈𝜃(0). Since 𝐺 is 𝑝⁻¹-open, there is 𝜀>0 such that 𝐵_𝑝−1(𝑥,𝜀)⊂𝐺. Since 𝑝⁻¹(𝑦−𝑥)=𝑝(𝑥−𝑦)=0, we have that (𝑦−𝑥)∈𝐵_𝑝−1(0,𝜀), then 𝑦∈𝑥+𝐵_𝑝−1(0,𝜀)=𝐵_𝑝−1(𝑥,𝜀)⊂𝐺.
(3) Suppose that 𝑦∈𝐹+𝜃(0). Then, 𝑦=𝑥+𝑧 with 𝑥∈𝐹 and 𝑧∈𝜃(0). Suppose that 𝑦∉𝐹. Since 𝑋⧵𝐹 is 𝑝⁻¹-open, by (2) we have that 𝑦−𝑧=𝑥∈𝑋⧵𝐹 and this yields a contradiction.
(4) Since 𝑋 is right-bounded there is 𝑟>0 such that 𝐵_𝑝(0,𝑟)⊂𝐵_𝑝𝑠(0,1)+𝜃(0), then 𝛿𝐵_𝑝(0,𝑟)⊂𝛿(𝐵_𝑝𝑠(0,1)+𝜃(0))⊂𝐵_𝑝𝑠(0,𝛿)+𝜃(0). Thus, if 𝑘=𝑟𝛿 then 𝐵_𝑝(𝑎,𝑘)=𝑎+𝐵_𝑝(0,𝑘)⊂𝑎+𝐵_𝑝𝑠(0,𝛿)+𝜃(0)=𝐵_𝑝𝑠(𝑎,𝛿)+𝜃(0). (2.11)

Theorem 2.11. If (𝑋,𝑝) is a biBanach right-bounded asymmetric normed space, then (𝑋,𝑝) is of the half second category.

Proof. Suppose 𝑋=⋃^∞_𝑛=1𝐹_𝑛. Then 𝑋=⋃^∞_𝑛=1cl_𝑝−1𝐹_𝑛. Since cl_𝑝−1𝐹_𝑛 is 𝑝^𝑠-closed, for all 𝑛∈ℕ, and (𝑋,𝑝^𝑠) is of the second category, because it is a Banach space, there is 𝑛₀∈ℕ such that int_𝑝𝑠cl_𝑝−1𝐹_𝑛0≠∅. Therefore, there exist 𝑎∈cl_𝑝−1𝐹_𝑛0 and 𝑡>0 such that 𝐵_𝑝𝑠(𝑎,𝑡)⊂𝑐𝑙_𝑝−1𝐹_𝑛0. Since 𝑋 is right bounded, by (4) of Lemma 2.10, there exists 𝑘>0 such that 𝐵_𝑝(𝑎,𝑘)⊂𝐵_𝑝𝑠(𝑎,𝑡)+𝜃(0). Therefore, by (3) of Lemma 2.10, we have that 𝐵_𝑝(𝑎,𝑘)⊂𝐵_𝑝𝑠(𝑎,𝑡)+𝜃(0)⊂cl_𝑝−1𝐹_𝑛0+𝜃(0)⊂cl_𝑝−1𝐹_𝑛0. (2.12)

The following result is immediate by Remark 2.9.

Corollary 2.12. If (𝑋,‖⋅‖) is a Banach lattice and 𝑝(𝑥)=‖𝑥⁺‖, then (𝑋,𝑝) is of the half second category.

Lemma 2.13. Let (𝑋,𝑝) be a right-bounded asymmetric normed space such that 𝜃(0) has a nonempty 𝑝^𝑠-interior. If 𝐹 is 𝑝⁻¹-closed, then the 𝑝-interior of 𝐹 is nonempty.

Proof. Since int_𝑝𝑠𝜃(0)≠∅, there exist 𝑎∈𝜃(0) and 𝛿>0 such that the ball 𝐵_𝑝𝑠(𝑎,𝛿)⊂𝜃(0). Since (𝑋,𝑝) is right-bounded, by (4) of Lemma 2.10, there is 𝑟>0 such that 𝐵_𝑝(𝑎,𝑟)⊂𝐵_𝑝𝑠(𝑎,1)+𝜃(0). Then 𝐵_𝑝()⊂1𝑎,𝑟𝛿𝐵_𝑝𝑠(𝑎,𝛿)+𝜃(0)⊂𝜃(0)+𝜃(0)=𝜃(0). (2.13)
If 𝑥∈𝐹, by (3) of Lemma 2.10, 𝑥+𝜃(0)⊂𝐹, and so 𝑥+𝐵_𝑝(𝑎,𝑟)⊂𝑥+𝜃(0)⊂𝐹. (2.14)

Hence, int_𝑝𝐹≠∅.

The following proposition is an immediate consequence of Lemma 2.13.

Proposition 2.14. Let (𝑋,𝑝) be a right-bounded asymmetric normed space. If 𝜃(0) has a nonempty 𝑝^𝑠-interior, then (𝑋,𝑝) is of the half second category.

Note that in the class of right-bounded asymmetric normed spaces there are spaces with empty 𝑝^𝑠-interior of 𝜃(0). In fact, if (𝑋,‖⋅‖) is an AL-space (Banach normed lattice where ‖𝑥∨𝑦‖=‖𝑥‖∨‖𝑦‖), then its positive cone 𝑃={𝑥∈𝑋∶𝑥≥0} has empty interior (see [1, page 357]). If we consider the canonical asymmetric normed space (𝑋,𝑝), then 𝜃(0)=−𝑃 and so 𝜃(0) has empty 𝑝^𝑠-interior since the norm 𝑝^𝑠 is equivalent to the original norm of 𝑋.

Acknowledgments

The authors are very grateful to the referee for many observations and comments that have allowed to improve the quality of the paper. The first two authors acknowledge the support of the Spanish Ministry of Science and Innovation under Grant MTM2009-12872-C02-01. The first author also acknowledges the support of the Spanish Ministry of Science and Innovation under Grant MTM2009-14483-C02-02.