# The Near-Ring of Lipschitz Functions on  a Metric Space

Authors: Mark Farag; Brink van der Merwe
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

This paper treats near-rings of zero-preserving Lipschitz functions on metric spaces that are also abelian groups, using pointwise addition of functions as addition and composition of functions as multiplication. We identify a condition on the metric ensuring that the set of all such Lipschitz functions is a near-ring, and we investigate the complications that arise from the lack of left distributivity in the resulting right near-ring. We study the behavior of the set of invertible Lipschitz functions, and we initiate an investigation into the ideal structure of normed near-rings of Lipschitz functions. Examples are given to illustrate the results and to demonstrate the limits of the theory.


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## Body

## 1. Introduction and Background

Banach spaces and Banach algebras of scalar-valued Lipschitz functions on a metric space have been studied in some depth by functional analysts for the past half of a century. The papers of Arens and Eells [1], de Leeuw [2], Sherbert [3, 4], and Johnson [5] contain some of the important early work on these topics. The book of Weaver [6] provides a systematic treatment of both the analytic and algebraic results concerning spaces of scalar-valued Lipschitz functions on a metric space. The Lipschitz functions considered therein are usually bounded and map a metric space (X,ρ) to a Banach space E (often ℝ or ℂ), so that addition (and multiplication in case E=ℝ or E=ℂ) of functions is defined. The Lipschitz number of a function f, denoted ∥f∥L, is used in combination with the infinity norm ∥·∥∞ to produce a norm ∥·∥:=max(∥·∥L,∥·∥∞). If one identifies a distinguished basepoint in X, then ∥·∥L is used as a norm on the set of basepoint preserving Lipschitz functions mapping (X,ρ) to E.In this paper, we initiate an analogous study of zero-preserving Lipschitz functions on a metric space that is also an abelian group, using pointwise addition of functions as addition and function composition as multiplication. Our Lipschitz functions, therefore, map a metric space(X,ρ) to itself and may be regarded as a generalization of the bounded linear operators that are so important in analysis. Rather than taking X to be a Banach space, we only require (X,+) to be an abelian group. By restricting the possible metrics ρ on X, we ensure that the set of zero-preserving Lipschitz functions on X is a near-ring under pointwise addition of functions and function composition. Near-rings and near-algebras, the nonlinear counterparts of rings and algebras, respectively, have a rich theory of their own. Basic near-ring definitions and results can be found in the books of Pilz [7], Clay [8], and Meldrum [9]; the dissertation of Brown [10] is the seminal work in near-algebras. Closely related to our present work is the dissertation of Irish [11], which considers near-algebras of Lipschitz functions on a Banach space.In the next section, we give the required definitions and elementary results. Next, we study the behavior of the set of units and also of ideals in normed near-rings of Lipschitz functions under topological closure. We conclude the paper by investigating the ideal structure of near-rings of Lipschitz functions.


## 2. Definitions, Notation, and Elementary Results

We begin this section by recalling the definition of a Lipschitz function.Definition 2.1 (see [6, 12]).
A functionf from a metric space (X1,ρ1) to a metric space (X2,ρ2) is Lipschitz if there exists a constant K≥0 such that for all x,y∈X1, ρ2(f(x),f(y))≤Kρ1(x,y). If f:(X1,ρ1)→(X2,ρ2) is Lipschitz, the Lipschitz number of f is defined as
(2.1)∥f∥L:=sup{ρ2(f(x),f(y))ρ1(x,y)∣x,y∈X1,x≠y}.Remark 2.2.
Iff is Lipschitz, then ∥f∥L≥0. Also, ∥f∥L=0 if and only if f is constant. For Lipschitz functions f:(X1,ρ1)→(X2,ρ2) and g:(X2,ρ2)→(X3,ρ3), we have that ∥g∘f∥L≤∥g∥L∥f∥L.Remark 2.3.
It is well-known that a Lipschitz functionf:ℝ→ℝ is absolutely continuous and therefore differentiable almost everywhere (a.e.). Also, the derivative of f is bounded a.e. in magnitude by the Lipschitz constant, and for a≤b, the difference f(b)-f(a) is equal to the integral of the derivative of f on the interval [a,b]. Conversely, if f:ℝ→ℝ is absolutely continuous (and thus differentiable a.e.) and if |f′(x)|≤K a.e., then f is Lipschitz with Lipschitz constant at most K. We will only use these observations in the case where f is a continuous function from ℝ to ℝ that is everywhere, except possibly for finitely many points, differentiable; also, the derivative of f will be continuous at all points where it exists. In this case f is Lipschitz if and only if the set {|f'(x)|:f is differentiable at x} is bounded and
(2.2)∥f∥L=sup{|f′(x)|:fisdifferentiableatx}.
In this restricted case these facts about real-valued Lipschitz functions are elementary consequences of the definition of Lipschitz functions, the definition of derivatives of real-valued functions, and  the mean value theorem. The interested reader can consult [12] or [6] for more on Lipschitz functions.In what follows, all metric spaces(X,ρ) are also abelian groups under the operation +, with identity element 0. We exclude the trivial case X={0}. If the metric ρ satisfies the condition in the next definition, the pointwise addition of two Lipschitz functions is again a Lipschitz function.Definition 2.4.
LetK>0 be a real number. A metric ρ on a metric space X is K-subadditive on X if
(2.3)ρ(a+b,c+d)ρ(a,c)+ρ(b,d),
for all a,b,c,d∈X with ρ(a,c)+ρ(b,d)≠0, bounded, and
(2.4)supρ(a,c)+ρ(b,d)≠0ρ(a+b,c+d)ρ(a,c)+ρ(b,d)=K.Remark 2.5.
Ifρ is a metric on X and we define the metric ρ¯ on X×X via ρ¯((a,b),(c,d))=ρ(a,c)+ρ(b,d), then K-subadditivity of ρ is equivalent to having the Lipschitz number of +:X×X→X equal to K.Example 2.6.
Let(X,∥·∥) be a normed vector space and let the metric ρ be defined on X by ρ(x,y)=∥x-y∥ for all x,y∈X. Then ρ is 1-subadditive.Assume thatρ is a 1-subaddive metric on the abelian group X, and define ∥·∥:X→ℝ+∪{0} by ∥x∥:=ρ(x,0) for all x∈X. We will show that ∥·∥ satisfies the properties given in the next definition.Definition 2.7 (see, e.g., [13]).
A function∥·∥:X→ℝ+∪{0} is a norm on the abelian group X if ∥·∥ satisfies the following criteria:  (1)
∥x∥=0 if and only if x=0;(2)
∥x∥=∥-x∥ for all x∈X;(3)
∥x+y∥≤∥x∥+∥y∥ for all x,y∈X.Remark 2.8.
Let∥·∥ be a norm on an abelian group. Then the function ρ:X×X→ℝ+∪{0}, defined by ρ(x,y):=∥x-y∥ for all x,y∈X, is a 1-subadditive metric on X.Example 2.9.
LetX be a multiplicative subgroup of the unit circle in the complex plane, and take “addition” in X to be complex multiplication (so that 1∈X is the neutral element). If we denote by ∥z∥ the Euclidean distance between the complex numbers z and 1, ∥·∥ is a norm on the abelian group X.Next we give some of the elementary properties ofK-subadditive metrics.Proposition 2.10.
Assume thatρ is K-subadditive on the metric space X. Then(1)
ρ(x-y,0)≤Kρ(x,y) and ρ(x,y)≤Kρ(x-y,0);(2)
ρ(x,y)≤K2ρ(-x,-y) and ρ(-x,-y)≤K2ρ(x,y);(3)
K≥1;(4)
ifK=1, then ρ(x,y)=ρ(x-y,0) and ρ(x,y)=ρ(-x,-y);(5)
ifK=1, then ∥·∥:X→ℝ+∪{0}, defined by ∥x∥:=ρ(x,0) for all x∈X, is a norm on X and ∥x-y∥=ρ(x,y) for all x,y∈X.Proof.
We prove (1) and (2). The other parts follow immediately from the following:
(1)
(2.5)ρ(x-y,0)=ρ(x-y,y-y)≤K(ρ(x,y)+ρ(-y,-y))=Kρ(x,y),
and
(2.6)ρ(x,y)=ρ((x-y)+y,0+y)≤K(ρ(x-y,0)+ρ(y,y))=Kρ(x-y,0).
(2)
(2.7)ρ(x,y)=ρ(y,x)=ρ(x-x+y,x-y+y)≤K[ρ(x-x,x-y)+ρ(y,y)]≤K2[ρ(x,x)+ρ(-x,-y)]=K2ρ(-x,-y).
Thus ρ(x,y)≤K2ρ(-x,-y) and therefore also ρ(-x,-y)≤K2ρ(x,y).Remark 2.11.
Note that from part (5) of Proposition 2.10 and Remark 2.8 we have that any 1-subadditive metric on an abelian group X is induced by a norm on X and, conversely, any norm on X induces a 1-subadditive metric.The following is an example of a metric space with a metric that is notK-subadditive for any K.Example 2.12.
Consider the metric onℝ given by ρ(a,b):=|2a-2b| for a,b∈ℝ. Then if ρ is K-subadditive, we would have that 2n-1=ρ(n,n-1)≤K for all n∈ℕ, which is clearly not possible.Notation 2.13.
For a metric space(X,ρ), we denote by ℒX,ρ, or simply ℒX when there is no ambiguity about the metric, the set of zero-preserving Lipschitz functions on X.Using some of the elementary properties ofK-subadditive metrics, we obtain the following properties for ∥·∥L.Proposition 2.14.
Letρ be a K-subadditive metric on X. Then for all f,g∈ℒX one has (1)
∥f∥L≥0 and ∥f∥L=0 if and only if f=0X;(2)
∥-f∥L≤K2∥f∥L;(3)
∥f∘g∥L≤∥f∥L∥g∥L;(4)
∥f+g∥L≤K(∥f∥L+∥g∥L).Proof.
The result follows from Definitions2.1 and 2.4 and Proposition 2.10.We now recall the definition of a near-ring.Definition 2.15.
A triple (N,+,*) is called a (right) near-ring if  (1)
(N,+) is a (not necessarily abelian) group,(2)
(N,*) is a semigroup,(3)
for alla,b,c∈N, (a+b)*c=a*c+b*c.A near-ring (N,+,*) is called zero-symmetric if, for all n∈N, n*0N=0N, where 0N is the neutral element of (N,+).
If(G,+) is any group, then ℳ(G), the set of all self-maps of G, is a near-ring under pointwise addition and function composition. The set of all zero-preserving self-maps of G, ℳ0(G), is a zero-symmetric sub-near-ring of ℳ(G). Further examples of near-rings, along with many of the basic results of the theory of near-rings, may be found in the books of Clay [8], Meldrum [9], and Pilz [7].Following the definition of a normed ring as given in [14], we make the following analogous definition.Definition 2.16.
Anormed near-ring(N,∥·∥) is a near-ring N with a function ∥·∥:N→ℝ+∪{0}, such that(1)
∥x∥=0 if and only if x=0;(2)
∥x∥=∥-x∥ for all x∈N;(3)
∥xy∥≤∥x∥∥y∥ for all x,y∈N;(4)
∥x+y∥≤∥x∥+∥y∥ for all x,y∈N.Proposition 2.17.
Assume thatρ is a K-subadditive metric on X. Then(1)
(ℒX,ρ,+,∘) (“+” is pointwise addition and “∘” is function composition) is a zero-symmetric near-ring with identity;(2)
(ℒX,ρ,∥·∥L) is a normed near-ring if K=1.Proof.
(1) We show that if f,g∈ℒX, then f∘g,f+g,-f∈ℒX. From Remark 2.2, f∘g∈ℒX, and from Proposition 2.10, we have for x,y∈X,
(2.8)ρ(-f(x),-f(y))≤K2ρ(f(x),f(y)),
and thus f∈ℒX implies that -f∈ℒX. Also,
(2.9)ρ(f(x)+g(x),f(y)+g(y))≤Kρ(f(x),f(y))+Kρ(g(x),g(y)),
since ρ is K-subadditive. Thus f,g∈ℒX implies that f+g∈ℒX. Finally, note that since ℒX contains only zero-preserving functions, it is a zero-symmetric near-ring.
(2) This result follows from (1) and Proposition 2.14.In some of our results, we assume that our metric space is a normed vector space over a field with an absolute value or norm. Recall that an absolute value on a fieldF is a function |·|:F→ℝ+∪{0}, such that(1)
|x|=0 if and only if x=0;(2)
|xy|=|x||y| for all x,y∈F;(3)
|x+y|≤|x|+|y| for all x,y∈F.IfX is a normed vector space over a normed field, ℒX is not only a normed near-ring but in fact a normed near-algebra. First we recall the definitions of a near-algebra and a normed near-algebra. We only consider near-algebras with identity.Definition 2.18 (see [7]).
A vector spaceA over a field F together with another binary operation “·” is a (right) near-algebra over F if (A,+,·) is a (right) near-ring and for all a,b∈A and all k∈F, (ka)·b=k(a·b).As with near-rings, near-algebras need not be zero-symmetric in general, as seen in the following example [15].Example 2.19.
LetX be a vector space over a field F. Then ℳ(X), the set of all self-maps of X, is a near-algebra over F which is not zero-symmetric. Also, if R is any subalgebra of the F-algebra EndF(X), then the set of all affine transformations arising from elements of R and X, that is, 𝒜ℱR(X):={Av∣A∈R,v∈X}, where Av(x):=Ax+v, is a sub-near-algebra of ℳ(X) which is also not zero-symmetric.Definition 2.20 (see [11]).
Anormed near-algebra(A,∥·∥) over a field (F,|·|) with an absolute value |·| is a near-algebra such that (A,∥·∥) is also a normed vector space over (F,|·|), with the norm ∥·∥ satisfying ∥f·g∥≤∥f∥∥g∥ for all f,g∈A.Proposition 2.21.
Let(X,∥·∥) be a normed vector space over a normed field (F,|·|). Then ℒX is a normed near-algebra over F.Proof.
This result follows from Definition2.1 and Proposition 2.17.In the remainder of the paper, we use the induced topology obtained from∥·∥L when making topological statements about ℒX.


## 3. Units inℒX

We note for the reader that there is some overlap between this section and [11, Chapter 3]. However, our proofs are less involved and our results are more general since in [11] only the case where X is a Banach space over ℝ or ℂ is considered. In this section we show that if X is a complete and connected normed abelian group, then the set of units is an open subset of ℒX. Throughout this discussion, the metric on X will be the 1-subadditive metric induced by the norm on X, as described in Remark 2.11. We start with a technical lemma that is required for the proof of Lemma 3.2. We denote the identity function in ℒX by 1X.Lemma 3.1.
Let(X,∥·∥) be a normed abelian group. Also, let f∈ℒX, with ∥f-1X∥L<1, and x,y∈X with x≠y. Then
(3.1)0<1-∥f-1X∥L≤∥f(x)-f(y)∥∥x-y∥≤1+∥f-1X∥L.Proof.
The statement follows from the fact that we have for allx,y∈X with x≠y that
(3.2)∥f-1X∥L≥∥(f(x)-x)-(f(y)-y)∥∥x-y∥≥|∥f(x)-f(y)∥-∥x-y∥|∥x-y∥=|∥f(x)-f(y)∥∥x-y∥-1|.The next lemma shows that iff∈ℒX is surjective and “close enough” to 1X, then f is a unit in ℒX.Lemma 3.2.
Let(X,∥·∥) be a normed abelian group. Also, let f∈ℒX be such that ∥f-1X∥L<1. Then f is injective and ∥f-1∥L≤1/(1-∥f-1X∥L), where f-1:f(X)→X.Proof.
From Lemma3.1 it follows that if x≠y, but f(x)=f(y), then 0<1-∥f-1X∥L≤∥f(x)-f(y)∥/∥x-y∥=0, which is not possible. Thus f must be injective.
Note that(3.3)sup{∥f-1(x)-f-1(y)∥∥x-y∥:x,y∈f(X),x≠y}≤sup{∥x-y∥∥f(x)-f(y)∥:x,y∈X,x≠y}(3.4)≤11-∥f-1X∥L,
where (3.3) follows by replacing x and y by f-1(x) and f-1(y), respectively, in sup{∥x-y∥/∥f(x)-f(y)∥:x,y∈X,x≠y}, and (3.4) follows from Lemma 3.1. Thus f-1 is Lipschitz with ∥f-1∥L≤1/(1-∥f-1X∥L).The next two lemmas will be used in the proof of the main result of this section.Notation 3.3.
Fora∈X and r∈ℝ+, we denote by Br(a) the set {x∈X:∥x-a∥<r}. Also, for A⊆X, we denote by A¯ the topological closure of A.Lemma 3.4.
Let(X,∥·∥) be a normed abelian group. Assume f∈ℒX with ∥f-1X∥L<1. Then there is an α∈ℝ+ such that Bαr(f(a))⊆f(Br(a))¯ for all r∈ℝ+ and a∈X.Proof.
If∥f-1X∥L=0, we have that f=1X, and the result follows trivially. Thus we assume that 0<∥f-1X∥L<1. Choose any α with 0<α<1-∥f-1X∥L. Suppose that Bαr(f(a)) is not a subset of f(Br(a))¯, and let c∈Bαr(f(a))∖f(Br(a))¯. Also, let d(c,f(Br(a))):=inf{∥c-x∥:x∈f(Br(a))}>0. Choose b∈Br(a) with ∥c-f(b)∥<d(c,f(Br(a)))/∥f-1X∥L. Let d:=b+c-f(b). We show next that  (1)
d∈Br(a),and(2)
∥f(d)-c∥<d(c,f(Br(a))).Once we have (1) and (2), we have a contradiction with the definition of d(c,f(Br(a))), and we thus have that Bαr(f(a))⊆f(Br(a))¯. The fact that d∈Br(a) follows from the following:
(3.5)∥d-a∥=∥(f-1X)(a)-(f-1X)(b)+(c-f(a))∥≤∥f-1X∥L∥a-b∥+∥f(a)-c∥<∥f-1X∥Lr+αr<r.
The fact that ∥f(d)-c∥<d(c,f(Br(a))) follows from the following:
(3.6)∥f(d)-c∥=∥(f-1X)(b+c-f(b))-(f-1X)(b)∥≤∥f-1X∥L∥b+c-f(b)-b∥=∥f-1X∥L∥c-f(b)∥<d(c,f(Br(a))).Lemma 3.5.
Let(X,∥·∥) be a normed abelian group. Assume that X is complete, C is a closed subset of X, f∈ℒX, and ∥f-1X∥L<1. Then f(C) is closed in X.Proof.
Suppose thaty∈X and ∥f(xn)-y∥→0 as n→∞, where xn∈C for all n∈ℕ. From Lemma 3.2  f is injective and f-1:f(X)→X is Lipschitz. Note that 〈xn:n∈ℕ〉 is Cauchy since 〈f(xn):n∈ℕ〉 is Cauchy and since we have the following:
(3.7)∥xn-xm∥=∥f-1f(xn)-f-1f(xm)∥≤∥f-1∥L∥f(xn)-f(xm)∥.
Since X is complete and C is closed, 〈xn:n∈N〉 converges to, say, x∈C. Thus since f is continuous, f(x)=y, and we therefore have that f(C) is closed in X.We introduce the main theorem of this section.Theorem 3.6.
LetX be a complete and connected normed abelian group. Then the set of units is open in ℒX.Proof.
In order to obtain the result, we show that ifg∈ℒX is a unit, then all f∈ℒX, with ∥f-g∥L<1/∥g-1∥L, are also units. First note that ∥f-g∥L<1/∥g-1∥L implies that ∥f∘g-1-1X∥L=∥(f-g)∘g-1∥L≤∥f-g∥L∥g-1∥L<1. Since f is a unit in ℒX if and only if f∘g-1 is a unit in ℒX, it is enough to show that ∥f-1X∥L<1 implies that f is a unit in ℒX. So assume that ∥f-1X∥L<1. From Lemma 3.2, f is injective. Lemma 3.5 implies that f(X) is closed, thus f(X)¯=f(X). Also, Lemma 3.4 implies that the set f(X)¯, which is equal to f(X), is open. Now since X is connected, we have that f(X) both open and closed implies that f(X)=X and thus that f is surjective. To complete the argument, we recall that Lemma 3.2 implies that f-1 is Lipschitz.We conclude this section by giving an example to show that the completeness ofX is an essential hypothesis in the preceding theorem.Example 3.7.
Definefn:ℚ→ℚ as follows:
(3.8)fn(x)={xx<1-1n,x22+(1-1n)-(1-1/n)22x∈[1-1n,1],x-12+(1-1n)-(1-1/n)22x>1.
Let f¯n:ℝ→ℝ be the continuous function such that the restriction of f¯n to ℚ is equal to fn. It follows from Remark 2.3, by calculating piecewise derivatives, that fn:ℚ→ℚ is Lipschitz for n>1. Let kn be a rational number such that 1-1/n<kn2<1. Then f¯n(kn2) is rational but not in the range of fn. Therefore fn is not surjective and thus not a unit in ℒℚ for any n. Next we show that fn converges to 1ℚ, the identity in ℒℚ, and thus the set of units in ℒℚ is not open. Let 1ℝ be the identity function on ℝ. We show that fn converges to 1ℚ, by showing that ∥f¯n-1ℝ∥L converges to 0. From Remark 2.3 we have that it is enough to show that the absolute value of the derivative of hn=f¯n-1ℝ is bounded (where it is defined) by a constant Mn, where Mn→0 as n→∞. This is clearly the case on (-∞,1-1/n) and on (1,∞). But also on (1-1/n,1) the derivative of hn is x-1; so on (1-1/n,1) the absolute value of the derivative of hn is bounded by 1/n. Thus we conclude that the required constants Mn exist.


## 4. Continuity of Multiplication and Closure of Ideals

In the first example in this section, we show that iff,g,gn∈ℒX for all n∈ℕ with gn→g in ℒX as n→∞, then it is not necessarily the case that f∘gn converges to f∘g in ℒX. Since (4.1)∥gn∘f-g∘f∥N=∥(gn-g)∘f∥N≤∥gn-g∥N∥f∥N
in any normed (right) near-ring (N,∥·∥N), we have that if f,g,gn∈N for all n∈ℕ with gn→g in N as n→∞, then gn∘f converges to g∘f. Thus right multiplication is a continuous function in a normed near-ring, but left multiplication is not. An example, similar to the next example, but more involved, is given in [11].Example 4.1.
In this example we show that it is not necessarily the case that∥f∘g-f∘h∥L≤∥f∥L∥g-h∥L for f,g,h∈ℒX, and also it is not the case that if gn converges to g as n approaches infinity, then f∘gn converges to f∘g.
LetX=ℝ be endowed with the Euclidean metric d, so that ℒX is a normed near-algebra, and define f,g,h:ℝ→ℝ as follows:
(4.2)f(x):=d(x,[-1,1]);g(x):=x;h(x):=kx,withk>1fixed.
From Remark 2.3, ∥f∘g-f∘h∥L=k, whereas ∥f∥L=1 and ∥g-h∥L=k-1.
For eachn∈ℕ, let gn(x):=(1+1/n)x. Then by replacing h with gn and k by (1+1/n), we obtain that ∥f∘g-f∘gn∥L=1+1/n, whereas ∥f∥L=1 and ∥g-gn∥L=1/n. Thus gn converges to g, but it is not the case that f∘gn converges to f∘g.Notation 4.2.
LetI be a nonempty indexing set, and for i∈I, let Ai and Bi be nonempty subsets of X. Define ℒX(Ai→Bi:i∈I) by {f∈ℒX∣f(Ai)⊆Bi for i∈I}. If I is finite, we will use the notation ℒX(A1→B1,…,An→Bn).Remark 4.3.
Note that we use the notationℒX(S→{0}), instead of the familiar notation AnnℒX(S).The next proposition shows, for example, that the left ideal, obtained by considering the set of functions inℒX that annihilates a certain subset of X, is closed.Proposition 4.4.
Fori∈I, let Ai and Bi be nonempty subsets of the normed abelian group (X,∥·∥), with the Bi's closed. Then the set ℒX(Ai→Bi:i∈I) is a closed subset of ℒX.Proof.
Let〈fn:n∈ℕ〉 be a sequence in ℒX(Ai→Bi:i∈I), and let f∈ℒX be such that fn converges f. To show that ℒX(Ai→Bi:i∈I) is closed, we need to show that f∈ℒX(Ai→Bi:i∈I). Let ai∈Ai. Then fn(ai)∈Bi. Since ∥f(ai)-fn(ai)∥≤∥f-fn∥L∥ai∥ and ∥f-fn∥L converges to 0, we conclude that fn(ai) converges to f(ai). Since each Bi is closed, we conclude that f(ai)∈Bi, and thus f∈ℒX(Ai→Bi:i∈I).Theorem 4.5.
The closure of a right ideal of a normed near-ringN is again a right ideal of N.Proof.
Denote the norm onN by ∥·∥N. Let I⊆N be a right ideal and f,g in I¯ the closure of I. Assume that 〈fn〉 and 〈gn〉 are sequences in I converging to f and g, respectively. Then ∥(f-g)-(fn-gn)∥N≤∥f-fn∥N+∥g-gn∥N, and the right side of this inequality converges to 0. Thus (f-g)∈I¯. Next let h∈N. Then ∥fn∘h-f∘h∥N=∥(fn-f)∘h∥N≤∥fn-f∥N∥h∥N, and again the right side of the inequality converges to 0 as n→∞. Thus since fn∘h∈I for all n∈ℕ, we conclude that f∘h∈I¯. It follows that I¯ is a right ideal of N.Remark 4.6.
Recall from the previous section that the set of units is open inℒX if X is a complete, connected, normed abelian group. In such a case, if S is a proper subset  of ℒX that is closed under either left or right function composition by an arbitrary function in ℒX, then the closure of S will also be a proper subset of ℒX.


## 5. Ideals inℒX

This section contains some partial results on the ideal structure ofℒX. In the first example we show that ideals in ℒX are in abundance.Example 5.1.
Letℱ be a set of functions from ℝ0+ to ℝ0+. Denote by ℐℱX the set
(5.1){f∈ℒX:thereexistsF∈ℱsuchthat∥f(x)∥≤F(∥x∥)forx∈X}.
Assume that we have the following conditions on the functions in ℱ:  (i)
ifF,G∈ℱ, then there is an H∈ℱ with F(t)+G(t)≤H(t) for all t∈ℝ0+;(ii)
ifr∈ℝ0+ and F∈ℱ, then rF∈ℱ;(iii)
ifr∈ℝ0+ and F∈ℱ, then there is a G∈ℱ such that F(rt)≤G(t) for all t∈ℝ0+;(iv)
the functions inℱ are nondecreasing. We assume that X is a normed abelian group and show next (in part) that ℐℱX is an ideal in ℒX.
We show that ifg∈ℐℱX and f,h∈ℒX, then f∘(g+h)-f∘h∈ℐℱX. The other cases are handled similarly. Since g∈ℐℱX, there exists some G∈ℐℱX such that for all x∈X, ∥g(x)∥≤G(∥x∥). We need to show that there is an H∈ℐℱX, such that ∥(f∘(g+h)-f∘h)(x)∥≤H(∥x∥). For any x∈X we have
(5.2)∥f∘(g(x)+h(x))-f∘h(x)∥≤∥f∥L∥g(x)+h(x)-h(x)∥=∥f∥L∥g(x)∥≤∥f∥LG(∥x∥).
Thus ∥(f∘(g+h)-f∘h)(x)∥≤H(∥x∥) for H=∥f∥LG.In the next example we consider the set of bounded functions inℒX.Example 5.2.
LetX be a normed abelian group. In this example we consider ℬℒX, the set of bounded Lipschitz functions in ℒX. First we show that ℬℒX is a two-sided ideal. If ℱ consists of all bounded nondecreasing functions from ℝ0+ to ℝ0+, then ℐℱX=ℬℒX, and it thus follows from the previous example that ℬℒX is a two-sided ideal in ℒX. Next we consider the case when X=ℝ. We show that ℬℒℝ is not closed. Define for each n∈ℕ the Lipschitz function fn:ℝ→ℝ as follows:
(5.3)fn(x)={0x≤0,xx∈[0,1],xx∈[1,n2],nx>n2.
Let f:ℝ→ℝ be defined as follows:
(5.4)f(x)={0x≤0,xx∈[0,1],xx>1.
Then from Remark 2.3 it follows that ∥f-fn∥L→0 as n→∞. Thus we have a sequence of bounded functions that converges to an unbounded function, which implies that ℬℒℝ is not closed in ℒℝ.In the next few results we show that the Betsch-Wielandt density theorem for near-rings can be applied toℒX.Lemma 5.3.
Assume thatX is a vector space over a field containing ℝ. Let S⊆X with 0∈S. Fix x0 in X, and define d(S,x), for all x∈X, by d(S,x)=infs∈S∥s-x∥. Then the function fS,x0:X→X, defined by fS,x0(x):=d(S,x)x0 for all x∈X, is Lipschitz. Also fS,x0=fS¯,x0, where S¯ is the closure of S.Proof.
We show thatfS,x0 is a Lipschitz function and leave the proof of the equality fS,x0=fS¯,x0 to the reader. First note that it is easy to verify that |d(S,x)-d(S,y)|≤∥x-y∥. Thus
(5.5)∥fS,x0(x)-fS,x0(y)∥=∥d(S,x)x0-d(S,y)x0∥=|d(S,x)-d(S,y)|∥x0∥≤∥x-y∥∥x0∥,
and fS,x0 is therefore a Lipschitz function.We will use the next result to conclude that ifx,y∈X with x≠0, then there is an f∈ℒX such that f(x)=y.Corollary 5.4.
Assume thatX is a vector space over a field containing ℝ. Let x,y,z∈X with x nonzero and x≠z. Then there is an f∈ℒX such that f(x)=y and f(z)=0.Proof.
Note thatcf{0,z},y(x)=y and cf{0,z},y(z)=0 for an appropriate c∈F, where f{0,z},y is as in Lemma 5.3.Corollary 5.5.
Assume thatX is a vector space over a field containing ℝ. Then ℒX is not a ring.Proof.
Letx∈X∖{0} and let S={0,x}. Let fS,x be as in Lemma 5.3, and denote by 1X the identity function on X. Now note that we have that fS,x∘(1X+1X)(x)≠0=fS,x∘1X(x)+fS,x∘1X(x), since (x+x)∉S.Corollary 5.6.
Assume thatX is a vector space over a field containing ℝ. Then X is a type-2 primitive ℒX-module.Proof.
From Corollary5.4,  ℒXx=X for all x≠0. Also, fX=0 for f∈ℒX implies that f=0. Thus X is a 2-primitive ℒX-module.Corollary 5.7.
Assume thatX is a vector space over a field containing ℝ. Let a1,…,an and b1,…,bn be elements in X with distinct and nonzero ai's. Then there exists an f∈ℒX such that f(ai)=bi for i=1,…,n.Proof.
SinceX is a type-2 primitive ℒX-module, the Betsch-Wielandt density theorem for near-rings (see, e.g., [7]) can be applied when ℒX is not a ring. By the density theorem,  if a1,…,an and b1,…,bn are in X and the ai's are distinct and all nonzero, then there exists f∈ℒX such that f(ai)=bi.We conclude by exhibiting some of the maximal left ideals inℒX. With the exception of the statement that ℒ({x0}→{0}) is closed, the argument is solely based on the fact that X is a 2-primitive ℒX-module.Theorem 5.8.
Assume thatX is a vector space over a field containing ℝ. Let x0∈X with x0≠0. Then ℒ({x0}→{0}), often denoted by Annℒ({x0}), is a maximal closed left ideal that is not an ideal.Proof.
From Proposition4.4 we have that ℒX({x0}→{0}) is closed. It is easy to verify that ℒX({x0}→{0}) is a left ideal. Next we show that it is maximal. Assume that f∈ℒX∖ℒX({x0}→{0}). Then from Corollary 5.4 we can find a g∈ℒX such that g(f(x0))=x0. But then idX=g∘f+(idX-g∘f) and (idX-g∘f)∈ℒX({x0}→{0}),  implying that the left ideal generated by f and ℒX({x0}→{0}) is all of ℒX. It follows that ℒX({x0}→{0}) is a maximal left ideal.
Finally we show thatℒX({x0}→{0}) is not an ideal. Let 0,x be two distinct elements in X, with x≠x0. From Corollary 5.4 we have functions f,g∈ℒX with f(x)=x,f(x0)=0, and g(x0)=x. Then it follows that f∈ℒX({x0}→{0}), but f∘g∉ℒX({x0}→{0}), since we have f(g(x0))=x.



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*Source: 284875-2010-07-13.xml*