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Li et al. Journal of Inequalities and Applications 2011, 2011:128

http://www.journalofinequalitiesandapplications.com/content/2011/1/128

RESEARCH Open Access

Demi-linear duality

1* 1 2Ronglu Li , Aihong Chen and Shuhui Zhong

* Correspondence: rongluli@yahoo. Abstract

com.cn

1Department of Mathematics, As is well known, there exist non-locally convex spaces with trivial dual and therefore

Harbin Institute of Technology, the usual duality theory is invalid for this kind of spaces. In this article, for a 150001, P.R. China

topological vector space X, we study the family of continuous demi-linear functionalsFull list of author information is

available at the end of the article on X, which is called the demi-linear dual space of X. To be more precise, the spaces

with non-trivial demi-linear dual (for which the usual dual may be trivial) are

discussed and then many results on the usual duality theory are extended for the

demi-linear duality. Especially, a version of Alaoglu-Bourbaki theorem for the demi-

linear dual is established.

Keywords: demi-linear, duality, equicontinuous, Alaoglu-Bourbaki theorem

1 Introduction

Let ∈ { , } and X be a locally convex space over with the dual X’.Thereisa

beautifuldualitytheoryforthepair(X, X’) (see [[1], Chapter 8]). However, it is possi-

pble that X’ = {0} even for some Fréchet spaces such as L (0, 1) for 0 <p < 1. Then the

usual duality theory would be useless and hence every reasonable extension of X’ will

be interesting.

L (X,Y)Recently, , the family of demi-linear mappings between topological vectorγ,U

spaces X and Y is firstly introduced in [2].L (X,Y) is a meaningful extension of theγ,U

family of linear operators. The authors have established the equicontinuity theorem,

the uniform boundedness principle and the Banach-Steinhaus closure theorem for the

L (X,Y)extension . Especially, for demi-linear functionals on the spaces of test func-γ,U

tions, Ronglu Li et al have established a theory which is a natural generalization of the

usual theory of distributions in their unpublished paper “Li, R, Chung, J, Kim, D:

Demi-distributions, submitted”.

Let X,Y be topological vector spaces over the scalar field andN(X) the family of

neighborhoods of 0Î X. Let

C(0) = γ ∈ : limγ(t)= γ(0) = 0,| γ(t) |≥| t | if | t |≤ 1 .

t→0

Definition 1.1 [2, Definition 2.1] A mapping f: X® Y is said to be demi-linear if f(0)

=0 and there exists g Î C(0) and U ∈N(X) such that every x Î X, u ÎUand

t ∈ {t ∈ :| t |≤ 1} yield for which |r - 1| ≤ | g (t) |, |s| ≤ | g (t)| and f(x+tu)r,s ∈

= rf(x)+ sf(u).

© 2011 Li et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution

License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Li et al. Journal of Inequalities and Applications 2011, 2011:128 Page 2 of 15

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L (X,Y)We denote by the family of demi-linear mappings related to g Î C(0)γ,U

and U ∈N(X),andbyK (X,Y) the subfamily of L (X,Y) satisfying the follow-γ,U γ,U

ing property: if x Î X, u Î U and |t| ≤ 1, then f(x+tu)= rf(x)+ sf(u) for some s with |

s| ≤ | g (t)|. Let

(γ,U)X = f ∈L (X, ): f iscontinuous ,γ,U

(g, U)which is called the demi-linear dual space of X. Obviously, X’ ⊂ X .

In this article, first we discuss the spaces with non-trivial demi-linear dual, of which

the usual dual may be trivial. Second we obtain a list of conclusions on the demi-linear

(g, U) (g, U)

dual pair (X, X ). Especially, the Alaoglu-Bourbaki theorem for the pair (X, X )

is established. We will see that many results in the usual duality theory of (X, X’) can

(g, U)

be extended to (X, X ).

L (X,Y)Before we start, some existing conclusions about are given as follows. Inγ,U

general,L (X,Y) is a large extension of L(X, Y). For instance, if ||·||: X® [0, +∞)isγ,U

a norm, then · ∈ L (X, ) for every gÎ C(0). Moreover, we have the followingγ,X

Proposition 1.2 ([2, Theorem 2.1]) Let X be a non-trivial normed space, C>1, δ>0

and U ={u Î X:||u|| ≤ δ}, g(t)= Ct for . If Y is non-trivial, i.e.,Y ≠{0},thenthet ∈

L (X,Y)family of nonlinear mappings in is uncountable, and every non-zero linearγ,U

L (X,Y)operator T : X® Y produces uncountably many of nonlinear mappings in .γ,U

XDefinition 1.3Afamily Г ⊂ Y is said to be equicontinuous at x Î X if for every

W ∈N(Y), there exists V ∈N(X) such that f(x + V) ⊂ f(x)+Wforallf Î Г,and Г

is equicontinuous on X or, simply, equicontinuous if Г is equicontinuous at each xÎ X.

X

As usual, Г ⊂ Y is said to be pointwise bounded on X if {f(x): f Î Г}isboundedat

each x Î X,and f : X ® Y is said to be bounded if f(B) is bounded for every bounded

B ⊂ X.

The following results are substantial improvements of the equicontinuity theorem

and the uniform boundedness principle in linear analysis.

⊂L (X,Y)Theorem 1.4 ([2, Theorem 3.1]) If X is of second category and is aγ,U

pointwise bounded family of continuous demi-linear mappings, then Г is equicontinuous

on X.

Theorem 1.5 ([2, Theorem 3.3]) If x is of second category and ⊂L (X,Y) is aγ,U

pointwise bounded family of continuous demi-linear mappings, then Г is uniformly

bounded on each bounded subset of X, i.e.,{f(x): f Î Г, x Î B} is bounded for each B ⊂ X.

If, in addition, X is metrizable, then the continuity of fÎ Г can be replaced by bound-

edness of fÎ Г.

2 Spaces with non-trivial demi-linear dual

f ∈L (X, )Lemma 2.1 Let . For each xÎ X, uÎ U and |t| ≤ 1, we haveγ,U

| f(tu) |≤| γ(t) || f(u) |; (1)

| f(x+tu) −f(x) |≤| γ(t) | (| f(x) | + | f(u) |). (2)Li et al. Journal of Inequalities and Applications 2011, 2011:128 Page 3 of 15

http://www.journalofinequalitiesandapplications.com/content/2011/1/128

f ∈L (X, )Proof. Since , for each xÎ X, uÎ U and |t| ≤ 1, we have f(x + tu)= rfγ,U

(x)+ sf(u) where |r-1| ≤ |g(t)| and |s| ≤ |g(t)|. Then

| f(x+tu) −f(x) |=| (r −1)f(x)+sf(u) |≤| r −1 || f(x) | + | s || f(u) |≤| γ(t) | (| f(x) | + | f(u) |),

which implies (2). Then (1) holds by letting x = 0 in (2).

Theorem 2.2 Let X be a topological vector space and f : X ® [0, +∞)afunction

satisfying

(∗) f(0) = 0,f(−x)= f(x) and f(x+y) ≤ f(x)+f(y) whenever x,y ∈ X.

U ∈N(X)Then, for every g Î C(0) and , the following (I), (II), and (III) are equiva-

lent:

f ∈L (X, )(I) γ,U ;

(II) f(tu) ≤ |g(t)|f(u) whenever uÎ U and |t| ≤ 1;

f ∈K (X, )(III) .γ,U

Proof. (I)⇒ (II). By Lemma 2.1.

(II)⇒ (III). Let xÎ X, uÎ U and |t| ≤ 1. Then

f(x)−| γ(t) | f(u) ≤ f(x) −f(tu) ≤ f(x+tu) ≤ f(x)+f(tu) ≤ f(x)+ | γ(t) | f(u).

Define : [-|g(t)|, |g(t)|]®ℝ by (a)= f(x)+ af(u). Then is continuous and

ϕ(−| γ(t) |)= f(x)−| γ(t) | f(u) ≤ f(x+tu) ≤ f(x)+ | γ(t) | f(u)= ϕ(| γ(t) |).

So there is sÎ[-|g(t)|, |g(t)|] such that f(x + tu)= g(s)= f(x)+ sf(u).

K (X, ) ⊂L (X, )(III)⇒ (I). .γ,U γ,U

In the following Theorem 2.2, we want to know whether a paranorm on a topologi-

cal vector space X is inK (X, ) for some g and U. However, the following exampleγ,U

shows that this is invalid.

Example 2.3 Let ω be the space of all sequences with the paranorm||·||:

∞ 1 | x |j

x = ,∀x=(x ) ∈ ω.jj2 1+ | x |j

j=1

· ∈/L (ω, )Then, for every g Î C(0) and U ={u=(u): ||u|| < ε}, we have γ,U .ε j

· ∈/L (ω, )Otherwise, there exists gÎ C(0) and ε>0 such that and henceγ,U

1 1

u ≤| γ( ) | u ,for all u ∈ U and n ∈ε

n n

(N)1by Theorem 2.2. Pick N Î N with <ε. Let , ∀n Î N.N u =(0,··· ,0, n,0,···)2 n

1 n 1Then implies u Î U for each NÎN. It follows from u = < <εn N N n ε2 1+n 2

11 u 1 1 1 n 11+n 1nn| γ( ) |≥ =( )/( )= > ,∀n ∈ ,

N Nn u 2 1+1 2 1+n 2 n 2n

1that as n® ∞, which contradicts gÎ C(0).γ( ) 0nLi et al. Journal of Inequalities and Applications 2011, 2011:128 Page 4 of 15

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Note that the space ω in Example 2.3 has a Schauder basis. The following corollary

shows that the set of nonlinear demi-linear continuous functionals on a Hausdorff

topological vector space with a Schauder basis has an uncountable cardinality.

Corollary 2.4 Let X be a Hausdorff topological vector space with a Schauder basis.

Then for every gÎ C(0) and U ∈N(X), the demi-linear dual

(γ,U) is uncountable.X = f ∈L (X,R): f is continuousγ,U

Proof. Let {b } be a Schauder basis of X.Thereisafamily P of non-zero paranormsk

on X such that the vector topology on X is just sP, i.e., x ® x in X if and only if ||xa a

- x||® 0 for each ||·||Î P ([[1], p.55]).

∞ ∞Pick ||·|| Î P.Then s b =0 for some s b ∈ X and hencek k k kk=1 k=1

s b =0 for some k ÎN. For non-zero , define f : X® [0, +∞)byk k 0 c ∈ c0 0

∞

f ( r b )=| cr | s b .c k k k k k0 0 0

k=1

Obviously, f is continuous and satisfies the condition (*) in Theorem 2.2. Let g Î Cc

∞

(0), r b ∈ X and |t| ≤ 1. Thenk kk=1

∞ ∞ ∞

f (t r b )=| ctr | s b =| t || cr | s b =| t | f ( r b ) ≤| γ(t) | f ( r b )c k k k k k k k k c k k c k k0 0 0 0 0 0

k=1 k=1 k=1

f ∈K (X, ) ⊂L (X, ) U ∈N(X)and hence c γ,U γ,U for all by Theorem 2.2. Thus,

(γ,U)f :0 = c ∈ ⊂ X for all gÎ C(0) and U ∈N(X).c

Example 2.5 As in Example 2.3, the space (ω, ||·||) is a Hausdorff topological vector

(n)

space with the Schauder base e =(0,··· ,0, ,0,···): n ∈ .Definef : ω ® ℝ1n c,n

with f (u)=|cu | where u=(u)Î ω. Then we havec,n n j

(γ,U)f :0 = c ∈ ,n ∈ ⊂ ω = f ∈L (ω, ): f is continuousc,n γ,U

for every gÎ C(0) and U ∈N(ω)by Corollary 2.4.

Recall that a p-seminorm ||·|| (0 <p ≤ 1) on a vector space E is characterized by ||x||

p

≥ 0, ||tx|| = |t| ||x|| and ||x + y|| ≤ ||x|| + ||y|| for all and x, yÎ E. If, in addi-t ∈

tion, ||x|| = 0 implies x = 0, then, ||·|| is called a p-norm on E.

Definition 2.6 ([[3], p. 11][[4], Sec. 2]) A topological vector space X is semiconvex if

and only if there is a family {p } of (continuous) k -seminorms (0<k ≤ 1)suchthata a a

the sets {xÎ X : p (x)<1} form a neighborhood basis at 0, that is,a

1

x : p (x) < : p ∈ P,n ∈ Nα α

n

is a base of N(X), where P is the family of all continuous p-seminorms with

0<p ≤ 1.

A topological vector space X is locally bounded if and only if its topology is given by

a p-norm (0 <p ≤ 1) ([[5], §15, Sec. 10]).

Clearly, locally bounded spaces and locally convex spaces are both semiconvex.

Li et al. Journal of Inequalities and Applications 2011, 2011:128 Page 5 of 15

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Corollary 2.7 Let X be a semiconvex Hausdorff topological vector space and p a con-0

U = {x ∈ X : p (x) ≤ 1}∈N(X)tinuous k -seminorm (0<k ≤ 1)onX.Thenfor and0 0 0 0

k0 , the demi-linear dualγ(·)= e|·| ∈

(γ,U )0X = f ∈L (X, ):fiscontinuousγ,U0

p (·) (γ,U )0 0is uncountable. Especially, p (·), sin(p (·)),e −1 ⊂ X .0 0

Proof.Let P be the family of all continuous k -seminorms with 0 <k ≤ 1. Obviously,a a

the functionals in P satisfy the condition (*) in Theorem 2.2. Moreover, for each p Îa

P with k ≥ k , we havea 0

k kα 0cp (tx)= c | t| p (x) ≤ c | t| p (x)≤| γ(t) | cp (x),for all x ∈ X,| t |≤1and c ∈ ,α α α α

(γ,U )0and hence by Theorem 2.2.{cp : c ∈ ,k ≥ k }⊂ Xα α 0

Define f : X ® ℝ by f(x)=sin(p (x)), ∀x Î X. For each x Î X, u Î U and |t| ≤ 1,0 0

k k0 0there exists and θÎ [0,1] such thats ∈ [−| t| ,| t| ]

sin(p (x+tu)) = sin(p (x)+sp (u)) = sin(p (x))+cos(p (x)+ θsp (u))sp (u),0 0 0 0 0 0 0

i.e.,

p (u)0

f(x+tu)= f(x)+cos(p (x)+ θsp (u)) sf(u),0 0

sin(p (u))0

where

p (u) π0 k k0 0| cos(p (x)+ θsp (u)) s |≤ | t| ≤ e | t| = | γ(t) |,0 0

sin(p (u)) 20

(γ,U )0which implies that .f(·)=sin(p (·)) ∈ X0

p (x)0Define g : X ® ℝ by , ∀x ÎX. For each x Î X, u Î U and |t| ≤ 1,g(x)= e −1 0

k k0 0there exists s ∈ [−| t| ,| t| ] such that

sp (u)0e −1p (x+tu) p (x)+sp (u) sp (u) p (x) p (x)0 0 0 0 0 0e −1= e −1= e (e −1)+ (e −1),

p (x)0e −1

i.e.,

sp (u)0e −1sp (u)0g(x +tu)= e g(x)+ g(u).

p (x)0e −1

Then, there exists θ,hÎ [0,1] for which

sp (u) θsp (u) k0 0 0| e − 1 |=| e sp (u) |≤ e | s |≤ e| t | =| γ(t) |0

and

sp (u) θsp (u)0 0e −1 e sp (u)0 θsp (u) k0 0| |=| |≤ e | s |≤ e | s |≤ e| t | =| γ(t) | .

p (x) ηp (u)0 0e −1 e p (u)0

p (·) (γ,U )0 0Thus, .g(·)= e −1 ∈ XLi et al. Journal of Inequalities and Applications 2011, 2011:128 Page 6 of 15

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pExample 2.8 For0<p<1,letL (0,1)bethespaceofequivalenceclassesofmeasur-

able functions on [0,1], with

1

p

f = | f(t) | dt < ∞.

0

p pThen (L (0,1), ||·||)’ = {0} ([[1], p.25]). However, L (0,1) is locally bounded and hence

psemiconvex. By Corollary 2.7, if U ={f:||f|| ≤ 1} and g(·) = e|·| Î C(0), then the0

p (γ,U )0demi-linear dual (L (0,1), · ) contains various non-zero functionals.

A conjecture is that each topological vector space has a nontrivial demi-linear dual

space. However, this is invalid, even for separable Fréchet space.

Example 2.9 LetM(0,1) be the space of equivalence classes of measurable functions

on [0,1], with

1 | f(t) |

f = dt.

1+ | f(t) |0

ThenM(0,1)is a separable Fréchet space with trivial dual. In fact, the demi-linear

M(0,1)dual space of is also trivial, that is,

(γ,U)(M(0,1), · ) = {0} for each γ ∈ C(0) and U ∈N(M(0,1)).

1

(γ,U) Let . Let N ÎN be such that implies f ÎUand |uu ∈ (M(0,1), · ) f ≤k

N

N k−1 k(f)| < 1. Given f ∈M(0,1), write where f =0 off [ , ]. Thenf = fk kk=1 N N

N N−1

u(f)= u( f )= u( f +f )k k N

k=1 k=1

N−1

= r u( f )+s u(f )N k N N

k=1

soN−2

= r r u( f )+r s u(f )+s u(f )N N−1 k N N−1 N−1 N N

k=1

= ···

= r ···r r u(f )+r ···r s u(f )+···N 3 2 1 N 3 2 2

+r s u(f )+s u(f ),N N−1 N−1 N N

N N−1

u(f)= u( f )= u( f +f )k k N

k=1 k=1

N−1

= r u( f )+s u(f )N k N N

k=1

N−2 (3)

= r r u( f )+r s u(f )+s u(f )N N−1 k N N−1 N−1 N N

k=1

= ···

= r ···r r u(f )+r ···r s u(f )+···N 3 2 1 N 3 2 2

+r s u(f )+s u(f ),N N−1 N−1 N N

Li et al. Journal of Inequalities and Applications 2011, 2011:128 Page 7 of 15

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where |r -1| ≤ |g(1)| and |s| ≤ |g(1)| for 2 ≤ I ≤ N. Theni i

N−1 N−2| u(f)|≤ (1+ | γ(1) |) | u(f ) | +(1+ | γ(1) |) | γ(1)|| u(f ) | +···1 2

(4)

+(1+ | γ(1) |) | γ(1)|| u(f ) | + | γ(1)|| u(f ) |N−1 N

N−1 N−2≤ (1 + | γ(1) |) +(1+ | γ(1) |) | γ(1) | +···

(5)

+(1 + | γ(1) |) | γ(1) | + | γ(1) |

N−1 (6)=2(1+ | γ(1) | ) −1.

1sup | u(f) | < +∞So . Since nf ≤ for each n Î N and 1 ≤ k ≤ N, wef∈M(0,1) k N

have {nf : nÎN, kÎN} ⊂ U. Then by Lemma 2.1,k

1 1 1

| u(f ) | =| u( (nf )) |≤| γ( ) || u(nf ) |≤| γ( ) | sup | u(f) |k k k (7)

n n n f∈M(0,1)

holds for all nÎN and 1 ≤ k ≤ N. Letting n® ∞, (7) implies u(f)=0 for 1 ≤ k ≤ N.k

Hence,|u(f)| = 0 by (4). Thus, u=0.

(g,U)3 Conclusions on the demi-linear dual pair (X, X )

N(X)Henceforth, X and Y are topological vector spaces over , is the family of

(g,U)neighborhoods of 0 Î X, and X is the family of continuous demi-linear functionals

inL (X, ). Recall that for usual dual pair (X, X’) and A ⊂ X, the polar of A, writtenγ,U

°

as A, is given by

◦ A = {f ∈ X : | f(x)|≤ 1,∀x ∈ A}.

(g,U)

In this article, for the demi-linear dual pair (X, X )and A ⊂ X,wedenotethe

?

polar of A by A , which is given by

• (γ,U)A = f ∈ X : | f(x)|≤ 1,∀x ∈ A .

(g,U)

Similarly, for S ⊂ X ,

•S = {x ∈ X : | f(x)|≤ 1,∀f ∈ S}.

f ∈L (X,Y)Lemma 3.1. Let γ,U . For every uÎ U and nÎN,

n−1

f(nu)= αf(u),where | α |≤ 2(1+ | γ(1) | ) −1.

Proof. It is similar to the proof of (3)-(6) in Example 2.9.

(g,U) •Lemma 3.2. Let S ⊂ X . If S is equicontinuous at 0Î X, then, S ∈N(X) and sup

|f(x)| < +∞ for every bounded B ⊂ X.fÎS,xÎB

Proof. Assume that S is equicontinuous at 0Î X. There is U ∈N(X) such that |f(x)|

? •< 1 for all fÎ S and xÎ V. Then V ⊂ S and hence S ∈N(X).

• 1 •Let B ⊂ X be bounded. Since S ∩U ∈N(X),wehave B ⊂ S ∩U for some m Î

m

N. Then for each fÎ S and xÎ B,

x x m−1| f(x) | =| f(m ) |= | α || f( ) |≤ | α |≤ 2(1 + | γ(1) | ) −1

m mLi et al. Journal of Inequalities and Applications 2011, 2011:128 Page 8 of 15

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m-1by Lemma 3.1. Hence, sup |f(x)| ≤ 2(1 + |g(1)|) -1<+∞.fÎS,xÎB

(g,U)Lemma 3.3. Let S ⊂ X . Then S is equicontinuous on X if and only if S is equicon-

tinuous at 0Î X.

Proof.Assumethat S is equicontinuous at 0 Î X.Thereis W ∈N(X) such that |f

(ω)| < 1 for all fÎ S and ωÎ W.

Let x Î X and ε > 0. By Lemma 3.2, sup |f(x)| = M<+∞. Observing lim g(t)f ÎS t ®0

δ ε| γ( ) |<=0,pick δ Î (0, 1) such that . By Lemma 2.1, for f Î S and2 2(M+1)

δ δu = u ∈ (W ∩U), we have02 2

δ δ ε

| f(x+u)−f(x) | =| f(x+ u ) −f(x) |≤| γ( ) | (| f(x) | + | f(u ) |) < (M+1)<ε.0 02 2 2(M+1)

δThus, f[x+ (W +U)] ⊂ f(x)+ {z ∈ : | z |<ε} for all f Î S, i.e., S is equicontinu-

2

ous at x.

(g,U) •Theorem 3.4. Let S ⊂ X . Then S is equicontinuous on X if and only if S ∈N(X).

•Proof.If S is equicontinuous, then S ∈N(X) by Lemma 3.2.

•Assume that S ∈N(X) and ε>0.Sincelim g(t)= g(0) = 0, there is δ>0sucht®0

δ δ •that |g(t)| <ε whenever |t|<δ. For f Î S and x = x ∈ (S ∩U),wehave|f(x )| ≤ 10 02 2

δ δand by Lemma 2.1. Thus,| f(x) | =| f( x ) |≤| γ( ) || f(x ) |<ε0 02 2

δ •f[ (S ∩U)]⊂{z ∈ : | z |<ε} for all f Î S, i.e., S is equicontinuous at 0 Î X.By2

Lemma 3.3, S is equicontinuous on X.

The following simple fact should be helpful for further discussions.

p

Example 3.5. Let (L (0, 1), ||·||) be as in Example 2.8, U={f:||f || ≤ 1} and g(t)= e

p p (g,U)

|t| for .Then (L (0, 1), ||·||) contains non-zero continuous functionals such ast ∈

||·|| p (g,U)

||·||, sin ||·||,e -1,etc.Since (af)(·) = af(·) for and f Î (L (0, 1), ||·||) ,itα ∈

||·|| p (g,U) 1 · p (γ,U)follows from e -1 Î (L (0, 1), ||·||) that (e − 1) ∈ (L (0,1), · ) . If u Î

e

1 u e−1U, then ||u|| ≤ 1, |sin ||u||| ≤ ||u|| ≤ 1 and .Thus,ifVisa| (e −1) |≤ < 1e e

p ?neighborhood of 0 Î L (0, 1) such that V ⊂ U, then V contains non-zero functionals

1 · such as ||·||, sin ||·||, (e − 1), etc.e

? (g,U)Corollary 3.6. For every U,V ∈N(X) and gÎ C(0),V ={fÎ X :|f(x)| ≤ 1, ∀xÎ

V} is equicontinuous on X.

? ? ? ? ?Proof.Let x Î V.Then|f(x)| ≤ 1, ∀f Î V , i.e., x Î (V ) . Thus, V ⊂ (V ) and so

?• •(V ) ∈N(X). By Theorem 3.4, V is equicontinuous on X.

(g,U)Corollary 3.7. If X is of second category and S ⊂ X is pointwise bounded on X,

•then S ∈N(X).

•Proof. By Theorem 1.4, S is equicontinuous on X. Then S ∈N(X) by Theorem 3.4.

(g,U)Corollary 3.8. Let X be a semiconvex space and S ⊂ X . Then S is equicontinuous

on x if and only if there exist finitely many continuous k-seminorm p ’s(0<k ≤ 1, 1 ≤ ii i i

≤ n<+∞) on x such that

sup sup | f(x) | < +∞. (8)

f∈S p (x)≤1,1≤i≤ni

In particular, for a p-seminormed space (X, ||·||) (||·|| is a p-seminorm for some p Î

(g,U)(0, 1], especially, a norm when p=1)andS ⊂ X , S is equicontinuous on x if and

Li et al. Journal of Inequalities and Applications 2011, 2011:128 Page 9 of 15

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only if

sup sup | f(x) | < +∞.

x ≤ 1f∈S

•S ∈N(X)Proof. Assume that S is equicontinuous. Then by Theorem 3.4. Accord-

ing to Definition 2.6, there exist finitely many continuous k-seminorm p ’s(0<k ≤ 1,i i i

1 ≤ i ≤ n<+∞) and ε > 0 such that

•{x ∈ X : p (x)<ε,1 ≤ i ≤ n}⊂ S ∩U.i

1 k0Let f Î S and p (x) ≤ 1, 1 ≤ i ≤ n.Pick n Î N for which ( ) <ε,where k =i 0 0n0

min k. Then1≤i≤n i

x 1 1k ki 0p ( )=( ) p (x) ≤ ( ) p (x)<ε,for1 ≤ i ≤ n,i i i

n n n0 0 0

x • x∈ S ∩U | f( ) |≤ 1which implies and hence . By Lemma 3.1,n n0 0

x x n −10| f(x) | =| f(n ) |=| αf( ) |≤ | α |≤ 2(1+ | γ(1) | ) −1.0

n n0 0

n −10sup sup | f(x)|≤ 2(1 + | γ(1) | ) −1 < +∞Thus, .f∈S p (x)≤1,1≤i≤ni

Conversely, suppose that p is a continuous k-seminorm with 0 <k ≤1for1 ≤ i ≤ ni i i

(g,U)1<+∞, and (8) holds. Let A = f : f ∈ S . Then A ⊂ X and

M+1

1 M

sup sup | g(x) | = sup sup | f(x) | = < 1,

1+M 1+Mg∈A p (x)≤1,1≤i≤n f∈S p (x)≤1,1≤i≤ni i

? ?•i.e., {x Î X : p (x) ≤ 1, 1 ≤ i ≤ n} ⊂ A and so A ∈N(X). By Theorem 3.4, A isi

equicontinuous on X and S=(1+ M)A is also equicontinuous on X.

X S ⊂ C(X, )Lemma 3.9. Let C(X, )= {f ∈ :fiscontinuous}.For ,thefollowing

(I) and (II) are equivalent.

(I) S is equicontinuous on X.

(II) If(x ) I is a net in x such that x ® x Î X, then lim f(x )= f(x) uniformly fora aÎ a a a

fÎ S.

Proof.(I)⇒(II). Let ε>0and x ® x in X.Since S is equicontinuous on X,thereisa

W ∈N(X) such that

| f(x+w) −f(x) |<ε,forall f ∈ S and w ∈ W.

Since x ® x, there is an index a such that x - xÎ W for all a ≥ a . Thena 0 a 0

| f(x ) −f(x) | = | f(x+x −x) −f(x) |<ε,forall f ∈ S andα>α .α α 0

Thus, lim f(x )= f(x) uniformly for fÎ S.a a

(II)⇒(I). Suppose that (II) holds but there exists x Î X such that S is not equicontin-

uous at x.

Then there exists ε > 0 such that for every V ∈N(X), we can choose f Î S and zv v

Î V for whichLi et al. Journal of Inequalities and Applications 2011, 2011:128 Page 10 of 15

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| f (x+z ) −f (x)|≥ ε (9)v v v

(x+z )Since (N(X),⊃) is a directed set, we have is a net in X. For everyv V∈N(X)

x+z ∈ x+V ⊂ x+W for all V ∈N(X)with W ⊃ V,,v

x+z ∈ x+V ⊂ x+W for all V ∈N(X)with W ⊃ V,v

that is, lim (x + z)= x.v v

By (II), there exists W ∈N(X) such that |f(x + z)- f(x)| <ε for all f Î S and0 v

V ∈N(X) with W ⊃ V.Then|f (x + z)- f (x)| <ε for all V ∈N(X) with W ⊃ V.0 v v v 0

This contradicts (9) established above. Therefore, (II) implies (I).

We also need the following generalization of the useful lemma on interchange of

limit operations due to E. H. Moore, whose proof is similar to the proof of Moore

lemma ([[6], p. 28]).

Lemma 3.10. Let D and D be directed sets, and suppose that D × D is directed by1 2 1 2

the relation (d ,d ) ≤ (d ,d ),whichisdefinedby d ≤ d and d ≤ d .Letf : D ×1 2 1 2 11 2 1 2

D ® X be a net in the complete topological vector space X. Suppose that:2

(a) for each d Î D , the limit g(d ) = lim f(d ,d )exists, and2 2 2 D 1 21

h(d ) = lim f(d ,d )(b) the limit exists uniformly on D .1 D 1 2 12

Then, the three limits

limg(d ),limh(d ), lim f(d ,d )2 1 1 2

D D D ×D2 1 1 2

all exist and are equal.

(g,U)We now establish the Alaoglu-Bourbaki theorem ([[1], p. 130]) for the pair (X, X ),

where X is an arbitrary non-trivial topological vector space.

XLet be the family of all scalar functions on X. With the pointwise operations (f +

Xg)(x)= f(x)+ g(x) and (tf)(x)=tf(x) for xÎ X and , we have is a lin-t ∈ x : →

Xear space and each x Î X defines a linear functional by letting x( f)= f(x)x : →

X Xfor . In fact, for and α,β ∈ ,f ∈ f,g ∈

x(αf + βg)=(αf + βg)(x)= αx(f)+ βx(g).

XThen, each xÎ X produces a vector topology ωx on such that

Xf → f in( ,ωx) if and only if f (x) → f(x)([1, p.12, p.38]).α α

XThe vector topology V {ωx : x Î X} is just the weak * topology in the pair (X, ),

Xand f ® f in ( ,weak∗) if and only if f (x) ® f(x) for each x Î X ( [[1], p. 12, p.a a

X38]). Note that weak* is a Hausdorff locally convex topology on .

(g,U) (g,Definition 3.11.AsubsetA ⊂ X is said to be weak * compact in the pair (X, X

U) X) or, simply, weak * compact if A is compact in ( ,weak∗), and A is said to be rela-

g,U

tively weak * compact in the pair (X, X ) or, simply, relatively weak* compact if in

X (γ,U)( ,weak∗) the closure ¯ is compact and ¯ .A A ⊂ X

(g,U) Xweak∗For A ⊂ X , ¯ stands for the closure of A in ( ,weak∗).A