`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 809626, 8 pageshttp://dx.doi.org/10.1155/2012/809626`
Research Article

## The Uniform Boundedness Theorem in Asymmetric Normed Spaces

1Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain
2Department of Mathematics, Indian Institute of Technology Madras, Chennai 6000 36, India

Received 5 July 2012; Accepted 27 August 2012

Copyright © 2012 C. Alegre et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 and 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 be a real vector space. A function is said to be an asymmetric norm on [2, 3] if for all , and ,(i) if and only if ;(ii);(iii).

The pair is called an asymmetric normed space.

Asymmetric norms are also called quasinorms in [46], and nonsymmetric norms in [7].

If is an asymmetric norm on , then the function defined on by is also an asymmetric norm on , called the conjugate of . The function defined on by is a norm on . We say that is a bi-Banach space if is a Banach space [3]. The following is a simple instance of a biBanach space.

Example 1.1. Denote by the function defined on by for all . Then is an asymmetric norm on such that is the Euclidean norm on , that is, is the Euclidean normed space . Hence is a biBanach space.

By a quasimetric on a nonempty set we mean a function such that for all (i) , and (ii) .

The function defined on by is a quasi-metric on called the conjugate of .

Each quasi-metric on induces a topology on which has as a base the family of the balls , where .

Each asymmetric norm on a real vector space induces a quasi-metric defined by , for all . We refer to the topology as the topology induced by . The terms -neighborhood, -open, -closed, and so forth will refer to the corresponding topological concepts with respect to that topology. The ball will be simply denoted by .

It was shown in [5] that for any normed lattice , the function defined on by , with , is an asymmetric norm on and the norm is equivalent to the norm . Moreover, determines both the topology and order of . We will refer to as the asymmetric norm associated to.

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, 810] 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, 1317]).

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 be a right K-complete asymmetric normed space, and let be an asymmetric normed space. If is a family of continuous linear operators from to such that for each , there exist and with and for all , then there exist and such that for all .

The condition of right K-completeness for 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 be a Banach space and let be a normed space. If is a family of continuous linear operators from to such that for each there exists with for all , then there exists such that for all .

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 be a biBanach asymmetric normed space and let be an asymmetric normed space. If is a family of continuous linear operators from to such that for each there exists with for all , then there exists such that for all .

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

Example 2.2. Let . Since is a closed linear subspace of for every , the subspace is a closed linear subspace of . We consider the asymmetric normed space , where
Since the norm is equivalent to the norm on , we have that is a closed linear subspace of the Banach space and then is a Banach space. Therefore is a biBanach space.
Let be given by for every . Since we have that is a continuous linear map from to , for every .
If , then , so that Therefore, , for every .
Now, we will prove that , for every . Indeed, if , then . If we consider such that , and if , then and . Hence,
Consequently,

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 is said to be of the first category if it is the union of a countable collection of nowhere dense subsets. 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 is of the second category if and only if condition implies for some .

If is a quasi-metric space and is a subset of , we denote by the closure of in the topological space and by the interior of in the topological space . If is an asymmetric normed space and is a subset of , we will write and instead of and , respectively.

Definition 2.3. We say that a quasi-metric space is of the half second category if , implies for some .

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

The quasi-metric space given in Example 1.1 is of the half second category. Indeed, if , take with . If , then . Otherwise, since nonempty proper -closed subsets of are of the form , there exists such that . Therefore, .

Note that is not of the second category. Indeed, in Proposition 1 of [20] it is proved that if is a normed lattice and , then is not of the second category.

Lemma 2.4 (uniform boundedness principle). If is a quasi-metric space of the half second category and is a family of real valued lower semicontinuous functions on such that for each there exists such that for all , then there exist a nonempty open set in and such that for all and .

Proof. For each let
Then each is closed in . Moreover . Indeed, by our hypothesis, given there exists such that for all , so .
Hence, there exists such that , where . Then, for each and we obtain . This completes the proof.

Definition 2.5. We say that an asymmetric normed space is of the half second category if the quasi-metric space is of the half second category.

Theorem 2.6 (uniform boundedness theorem). Let and be two asymmetric normed spaces such that is of the half second category. If is a family of continuous linear operators from to such that for each there exists with for all , then there exists such that for all .

Proof. For each define a function by for all . We first show that is lower semicontinuous on . Indeed, let and be a sequence in such that . Then . By the linear continuity of we deduce that . From the fact that for all , it follows that for each there is such that . Since is arbitrary, we conclude that is lower semicontinuous on .
Put . Since for each for all , we can apply Lemma 2.4 and thus there exists a nonempty -open subset of and a such that for all and .
Fix . Then, there exists such that . Take an . Then, , where . Put , and let such that . We will prove that for all . Indeed, first note that , so .
Now take . Then

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 be a Banach space and let be a normed space. If is a family of continuous linear operators from to such that for each there exists with for all , then there exists such that for all .

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

Definition 2.8. Let be an asymmetric normed space. is right bounded if there is such that , being .

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 is a normed lattice and , then the asymmetric normed space is right-bounded with constant .

Lemma 2.10. Let be an asymmetric normed space and let .(1)If is a -open subset of , then .(2)If is -open subset of , then .(3)If is a -closed subset of , then (4)If right bounded, then for all and for all , there is such that .

Proof. (1) Let . There exists such that . Since is -open, there is such that . Since , we have that , then .
(2) Let . There exists such that , that is, . Since is -open, there is such that . Since , we have that , then .
(3) Suppose that . Then, with and . Suppose that . Since is -open, by (2) we have that and this yields a contradiction.
(4) Since is right-bounded there is such that , then . Thus, if then

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

Proof. Suppose . Then . Since is -closed, for all , and is of the second category, because it is a Banach space, there is such that . Therefore, there exist and such that . Since is right bounded, by (4) of Lemma 2.10, there exists such that . Therefore, by (3) of Lemma 2.10, we have that

The following result is immediate by Remark 2.9.

Corollary 2.12. If is a Banach lattice and , then is of the half second category.

Lemma 2.13. Let be a right-bounded asymmetric normed space such that has a nonempty -interior. If is -closed, then the -interior of is nonempty.

Proof. Since , there exist and such that the ball . Since is right-bounded, by (4) of Lemma 2.10, there is such that . Then
If , by (3) of Lemma 2.10, , and so

Hence, .

The following proposition is an immediate consequence of Lemma 2.13.

Proposition 2.14. Let be a right-bounded asymmetric normed space. If has a nonempty -interior, then is of the half second category.

Note that in the class of right-bounded asymmetric normed spaces there are spaces with empty -interior of . In fact, if is an AL-space (Banach normed lattice where ), then its positive cone has empty interior (see [1, page 357]). If we consider the canonical asymmetric normed space , then and so has empty -interior since the norm is equivalent to the original norm of .

#### 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.

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