Demi-linear duality
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2011
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2011
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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usual theory of distributions in their unpublished paper “Li, R, Chung, J, Kim, D: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
http://www.journalofinequalitiesandapplications.com/content/2011/1/128
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
http://www.journalofinequalitiesandapplications.com/content/2011/1/128
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
