# The Uniform Boundedness Theorem in Asymmetric Normed Spaces

Authors: C. Alegre; S. Romaguera; P. Veeramani
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction

Throughout this paper the lettersℝ 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.LetX 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].Ifp 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 ps defined on X by ps(x)=max{p(x),p-1(x)} is a norm on X. We say that (X,p) is a bi-Banach space if (X,ps) is a Banach space [3]. The following is a simple instance of a biBanach space.Example 1.1.
Denote byu the function defined on ℝ by u(x)=x∨0 for all x∈ℝ. Then u is an asymmetric norm on ℝ such that us is the Euclidean norm on ℝ, that is, (ℝ,us) is the Euclidean normed space (ℝ,|·|). Hence  (ℝ,u)  is a biBanach space.By a quasimetric on a nonempty setX 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 functiond-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-metricd on X induces a T0 topology T(d) on X which has as a base the family of the balls {Bd(x,r):x∈X,r>0}, where Bd(x,r)={y∈X:d(x,y)<r}.Each asymmetric normp on a real vector space X induces a quasi-metric dp defined by dp(x,y)=p(y-x), for all x,y∈X. We refer to the topology T(dp) 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 Bdp(x,r) will be simply denoted by Bp(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 ps 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 bx>0 and cx>0 with q(f(x))≤bx and q-1(f(x))≤cx for all f∈ℱ, then there exist b>0 and c>0 such that
(1.1)sup{q(f(x)):p-1(x)≤1}≤b,sup{q-1(f(x)):p(x)≤1}≤c,
for all f∈ℱ.The condition of rightK-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]).
LetX 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 bx>0 with ∥f(x)∥≤bx 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.LetX 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 bx>0 with q(f(x))≤bx for all f∈ℱ, then there exists b>0 such that
(2.1)sup{q(f(x)):p(x)≤1}≤b,
for all f∈ℱ.Nevertheless, the following example shows that such a result does not hold in our context.Example 2.2.
LetXk={(xn)∈l1:2kx2k-1+x2k=0}. Since Xk is a closed linear subspace of l1 for every k∈ℕ, the subspace
(2.2)X=⋂k=1∞‍Xk,
is a closed linear subspace of l1. We consider the asymmetric normed space (l1,p), where
(2.3)p(x)=∥x+∥1=∑n=1∞‍xn+.
Since the normps is equivalent to the norm ∥·∥1 on l1, we have that X is a closed linear subspace of the Banach space (l1,ps) and then (X,ps) is a Banach space. Therefore (X,p) is a biBanach space.
Letfn:X→ℝ be given by fn(x)=(2n+1)x2n-1 for every n∈ℕ. Since
(2.4)fn(x)=(2n+1)x2n-1≤(2n+1)x2n-1+≤(2n+1)p(x),
we have that fn is a continuous linear map from (X,p) to (ℝ,u), for every n∈ℕ.
Ifx∈X, then ∥x∥1=∑n=1∞‍(2n+1)|x2n-1|, so that
(2.5)(2n+1)x2n-1≤(2n+1)|x2n-1|≤∥x∥1.
Therefore, fn(x)=(2n+1)x2n-1≤∥x∥1, for every n∈ℕ.
Now, we will prove thatsup{fn(x):p(x)≤1}=2n+1, for every n∈ℕ. Indeed, if p(x)≤1, then fn(x)≤2n+1. If we consider x=(xin)i=1∞ such that x2n-1n=1, x2nn=-2n and xin=0 if i∈ℕ-{2n-1,2n}, then x∈X,p(x)=1 and fn(x)=2n+1. Hence,
(2.6)sup{fn(x):p(x)≤1}=2n+1.
Consequently,(2.7)sup{sup{fn(x):p(x)≤1}:n∈ℕ}=∞.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 spaceX 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=1∞‍En implies int(clEm)≠∅ for some m∈ℕ.If(X,d) is a quasi-metric space and A is a subset of X, we denote by cld-1A the closure of A in the topological space (X,T(d-1)) and by intdA 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 clp-1A and intpA instead of cl(dp)-1A and intdpA, respectively.Definition 2.3.
We say that a quasi-metric space(X,d) is of the half second category if X=⋃n=1∞‍En, implies intd(cld-1Em)≠∅ for some m∈ℕ.Note that ifd 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,du) given in Example 1.1 is of the half second category. Indeed, if ℝ=⋃n=1∞‍En, take m∈ℕ with Em≠∅. If Em=ℝ, then cl(du)-1(intduEm)=ℝ. Otherwise, since nonempty proper (du)-1-closed subsets of ℝ are of the form ]-∞,a],a∈ℝ, there exists am∈ℝ such that cl(du)-1Em=]-∞,am]. Therefore, intdu(cl(du)-1Em)=]-∞,am[.Note that(ℝ,du) 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,dp) 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 bx>0 such that f(x)≤bx 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 eachn∈ℕ let
(2.8)En=⋂f∈ℱ‍f-1]-∞,n].
Then eachEn is closed in (X,d-1). Moreover X=⋃n=1∞‍En. Indeed, by our hypothesis, given x∈X there exists nx∈ℕ such that f(x)≤nx for all f∈ℱ, so x∈Enx.
Hence, there existsm∈ℕ such that U≠∅, where U=intd(cld-1Em)=intdEm. 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,dp) 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 bx>0 with q(f(x))≤bx for all f∈ℱ, then there exists b>0 such that
(2.9)sup{q(f(x)):p(x)≤1}≤b,
for all f∈ℱ.Proof.
For eachg∈ℱ define a function hg by hg(x)=q(g(x)) for all x∈X. We first show that hg is lower semicontinuous on (X,(dp)-1). Indeed, let x∈X and {xn}n∈ℕ be a sequence in X such that (dp)-1(x,xn)→0. Then p(x-xn)→0. By the linear continuity of g we deduce that q(g(x)-g(xn))→0. From the fact that q(g(x))-q(g(xn))≤q(g(x)-g(xn)) for all n∈ℕ, it follows that for each ɛ>0 there is n0∈ℕ such that hg(x)-hg(xn)<ɛ. Since x is arbitrary, we conclude that hg is lower semicontinuous on (X,(dp)-1).
Putℋ={hg:g∈ℱ}. Since for each x∈X,hg(x)=q(g(x))≤bx 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 hg(x)≤δ for all g∈ℱ and x∈U.
Fixz∈U. Then, there exists r>0 such that Bp(z,r)⊂U. Take an ɛ∈]0,r[. Then, B¯p(z,ɛ)⊂U, where B¯p(z,ɛ)={y∈X:p(y-z)≤ɛ}. Put b=(δ+b-z)/ɛ, 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(ɛx+z-z)=ɛp(x)≤ɛ, so ɛx+z∈U.
Now takef∈ℱ. Then
(2.10)q(f(x))=1ɛq(f(ɛx))=1ɛq(f(ɛx+z-z))=1ɛq(f(ɛx+z)+f(-z))≤1ɛ[q(f(ɛx+z))+q(f(-z))]=1ɛ[hf(ɛx+z)+hf(-z)]≤1ɛ(δ+b-z)=b.Since the Banach spaces are of the second category, the classical result for normed space is a corollary of this theorem.Corollary 2.7.
LetX 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 bx>0 with ∥f(x)∥≤bx 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 rBp(0,1)⊂Bps(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)
IfG is a p-open subset of X, then G+θ(0)⊂G.(2)
IfG is p-1-open subset of X, then G-θ(0)⊂G.(3)
IfF 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 Bp(a,k)⊂Bps(a,δ)+θ(0).Proof.
(1) Lety∈G+θ(0). There exists x∈G such that y-x∈θ(0). Since G is p-open, there is ɛ>0 such that Bp(x,ɛ)⊂G. Since p(y-x)=0, we have that (y-x)∈Bp(0,ɛ), then y∈x+Bp(0,ɛ)=Bp(x,ɛ)⊂G.
(2) Lety∈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 ɛ>0 such that Bp-1(x,ɛ)⊂G. Since p-1(y-x)=p(x-y)=0, we have that (y-x)∈Bp-1(0,ɛ), then y∈x+Bp-1(0,ɛ)=Bp-1(x,ɛ)⊂G.
(3) Suppose thaty∈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) SinceX is right-bounded there is r>0 such that Bp(0,r)⊂Bps(0,1)+θ(0), then δBp(0,r)⊂δ(Bps(0,1)+θ(0))⊂Bps(0,δ)+θ(0). Thus, if k=rδ then
(2.11)Bp(a,k)=a+Bp(0,k)⊂a+Bps(0,δ)+θ(0)=Bps(a,δ)+θ(0).Theorem 2.11.
If(X,p) is a biBanach right-bounded asymmetric normed space, then (X,p) is of the half second category.Proof.
SupposeX=⋃n=1∞‍Fn. Then X=⋃n=1∞‍clp-1Fn. Since clp-1Fn is ps-closed, for all n∈ℕ, and (X,ps) is of the second category, because it is a Banach space, there is n0∈ℕ such that intpsclp-1Fn0≠∅. Therefore, there exist a∈clp-1Fn0 and t>0 such that Bps(a,t)⊂clp-1Fn0. Since X is right bounded, by (4) of Lemma 2.10, there exists k>0 such that Bp(a,k)⊂Bps(a,t)+θ(0). Therefore, by (3) of Lemma 2.10, we have that
(2.12)Bp(a,k)⊂Bps(a,t)+θ(0)⊂clp-1Fn0+θ(0)⊂clp-1Fn0.The following result is immediate by Remark2.9.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 ps-interior. If F is p-1-closed, then the p-interior of F is nonempty.Proof.
Sinceintpsθ(0)≠∅, there exist a∈θ(0) and δ>0 such that the ball Bps(a,δ)⊂θ(0). Since (X,p) is right-bounded, by (4) of Lemma 2.10, there is r>0 such that Bp(a,r)⊂Bps(a,1)+θ(0). Then
(2.13)Bp(a,r)⊂(1δ)Bps(a,δ)+θ(0)⊂θ(0)+θ(0)=θ(0).
Ifx∈F, by (3) of Lemma 2.10, x+θ(0)⊂F, and so
(2.14)x+Bp(a,r)⊂x+θ(0)⊂F.Hence,intpF≠∅.The following proposition is an immediate consequence of Lemma2.13.Proposition 2.14.
Let(X,p) be a right-bounded asymmetric normed space. If θ(0) has a nonempty ps-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 emptyps-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 ps-interior since the norm ps is equivalent to the original norm of X.



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