Some results on random smoothness

R E S E A R C H

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Some results on random smoothness

Xiaolin Zeng

_{and Changying Ding}

*_{Correspondence:}

[email protected]

1_{School of Mathematics and}

Statistics, Chongqing Technology and Business University, Xuefu Road No. 21, Chongqing, 400067, P.R. China

Full list of author information is available at the end of the article

Abstract

Based on the analysis of stratification structure on random normed modules, we first present the notion of random smoothness in random normed modules. Then, we establish the relations between random smoothness and random strict convexity. Finally, a type of Gâteaux differentiability is defined for random norms, and its relation to random smoothness is given. The results are helpful in the further study of geometry of random normed modules.

Keywords: random normed module; random strict convexity; random smoothness; Gâteaux differentiability

1 Introduction

Guo initiated a new approach to random functional analysis [–], whose main idea is to develop random functional analysis as functional analysis based on random normed mod-ules, random inner product modules and random locally convex modules. In particular, since the essential applications of random functional analysis to conditional risk measures, the idea has also attracted much attention in different aspects [–]. Motivated by these advances, the study of geometric theory of random normed modules has begun in the direction of geometry of classical Banach spaces []. In [, ] random strict convexity and random uniform convexity are successfully introduced in random normed modules, which facilitates further study on geometrical properties of random normed modules []. The geometric theory of random normed modules is closely related to the geometry of Banach spaces since a complete random normed module is a kind of random generaliza-tion of a Banach space, as demonstrated by [, ]. From the point of view of classical Banach space theory, it is a quite natural topic to give a reasonable definition of random smoothness in random normed modules. Based on the analysis of stratification structure on random normed modules, we first present the notion of random smoothness via that of support functionals for the random closed unit ball in a random normed module. Then, the relations of random smoothness to random strict convexity are established. Finally, a type of Gâteaux differentiability equivalent to that in [] is introduced for random norms, and its relation to random smoothness is given.

The remainder of this article is organized as follows. Section  is devoted to some knowl-edge indispensable for next two sections. Section  is focused on the definition and basic properties of random smoothness, where Proposition . is of vital importance in this article. As a generalization of the corresponding classical case, Proposition . is a non-trivial result which plays a key role in the proof of Proposition .. Section  is focused on

the Gâteaux differentiability of random norm, where some inequality techniques are em-ployed in combination with stratification analysis in random normed modules to derive the main result Theorem ..

Throughout this paper, we adopt the following notations []:

(,F,P) denotes a probability space,K is the scalar fieldRof real numbers or C of complex numbers, L_{(F,K) denotes the algebra of equivalence classes ofK-valuedF}

-measurable random variables onin the usual sense.

For any A∈F,the equivalence class of A, denoted by A˜, is defined byA˜ ={B∈F :

P(AB) = }, whereABis the symmetric difference ofAandB.P(A˜) is defined to be

P(A). For twoF-measurable setsGandD,G⊂Da.s. meansP(G\D) = , in which case we also sayG˜ ⊂ ˜D;G˜ ∩ ˜Ddenotes the equivalence class determined byG∩Dand so on. For anyξ,η∈L_{(F,R),ξ >ηmeansξ ≥ηandξ =η. [ξ >η] stands for the equivalence}

class of theF-measurable set{ω∈:ξ(ω) >η(ω)}(briefly, [ξ>η]), whereξandη are arbitrarily chosen representatives ofξ andη, respectively, andI[ξ>η]stands forI˜[ξ>η].

Other analogous symbols are easily understood.

Specially,L₊={ξ∈L(F,R)|ξ≥}andFdenotes the set of equivalence classes of ele-ments inF.

2 Preliminaries

In this section, we recall Hahn-Banach theorems forL_{-linear functions and a.s. bounded}

random linear functionals, respectively, the notion of a random normed module together with its random conjugate space, the notion of random strict convexity and frequently-used notations.

LetE be a left module over the algebraL_{(F,K), a module homomorphismf :E→} L_{(F,K) is called anL}_(orL_{(F,K))-linear function. IfK=R, then a mappingp:E→} L_{(F,R) is called anL}_{-sublinear function if it satisfies the following:}

() p(ξx) =ξp(x),∀ξ∈L

+andx∈E;

() p(x+y)≤p(x) +p(y),∀x,y∈E.

Theorem . below is an important result which will be used in the proof of Theorem ..

Theorem .([]) Let E be a left module over the algebra L_{(F,R),M⊂E be an L}_(F,R)

-submodule,f :M→L_{(F,R)be an L}_{-linear function and p:E→L}_{(F,R)be an L}

-sublinear function such that f(x)≤p(x),∀x∈M.Then there exists an L-linear function

g:E→L_{(F,R)such that g extends f and g(x)≤p(x),∀x∈E.}

Definitions . and . below are fundamental notations well known in random metric theory.

Definition .([]) An ordered pair (E, · ) is called arandom normed module(briefly, anRNmodule) overKwith base (,F,P) ifEis a left module over the algebraL_(F,K)

and · is a mapping fromEtoL₊such that the following three axioms are satisfied:

(RNM-) ξx=|ξ|x,∀ξ∈L(F,K),x∈E; (RNM-) x+y ≤ x+y,∀x,y∈E;

(RNM-) x= impliesx=θ (the null vector inE),

Definition .([]) Let (E,·) be an RN module. A linear operatorffromEtoL_(F,K)

is analmost surely (briefly, a.s.) bounded random linear functionalonE if there exists some ξ inL

+ such that|f(x)| ≤ξ · x,∀x∈E. Denote byE∗ the linear space of all a.s.

bounded random linear functionals onEwith the ordinary pointwise addition and scalar multiplication on linear operators. Then (E∗, · ∗) is an RN module overK with base (,F,P), in which the module multiplication·:L_(F,K)×E∗_→E∗ _{is defined by (ξ ·}

f)(x) =ξ ·(f(x)),∀ξ∈L_{(F,K),f ∈E}∗ _{andx∈E, and the random norm ·} ∗_:E∗_→L +

is defined byf∗=∧{ξ ∈L

+:|f(x)| ≤ξ· x,∀x∈E},∀f ∈E∗. (E∗, · ∗) is calledthe

random conjugate spaceof (E, · ).

Remark . In Definition ., we can easily see that

() for eachf inE∗,|f(x)| ≤ f∗· x,∀x∈E;

() since(E∗, · ∗)is also an RN module, we can have its random conjugate space

((E∗)∗, ( · ∗)∗)(briefly,(E∗∗, · ∗∗)). Define a mappingJ:E→E∗∗by

J(x)(f) =f(x),∀f∈E∗andx∈E, thenJ(x)∗∗=x. Such a mappingJis calledthe canonical embedding mappingfromEtoE∗∗. In particular, whenJis surjective, we call(E, · )to berandom reflexive[].

Theorem . below is the Hahn-Banach theorem for a.s. bounded random linear func-tionals.

Theorem .([, ]) Let(E, · )be an RN module over K with base(,F,P),M⊂E

be a linear subspace,and f :M→L_{(F,K)be an a.s.bounded random linear functional}

on M.Then there exists F∈E∗such that()F(x) =f(x),∀x∈M,and()F∗=f∗.As a

consequence,for any x∈E,there exists g∈E∗such that g(x) =xandg∗=IAx.

Notations in Definition . below were heavily used in the study of random strict con-vexity and random uniform concon-vexity [] in order to analyze the stratification structure ofE.

Definition .([]) For anyx,yin an RN module (E, · ) overK with base (,F,P), [x = ] is denoted byAx, calledthe support of x, and we briefly writeAxy=Ax∩Ay

andBxy=Ax∩Ay∩Ax–y.The random unit sphere of Eis defined byS(E) ={x∈E:x=

˜

IAfor someA∈FwithP(A) > }.The random closed unit ball of Eis defined byU(E) = {x∈E:x ≤}.

Random strict convexity is as follows.

Definition .([]) An RN module (E, · ) is said to berandom strictly convexif for anyxandy∈E\{θ}such thatx+y=x+ythere existsξ∈L

+such thatξ>  onAxy andIAxyx=ξ(IAxyy).

It is well known that an RN module (E, · ) is random strictly convex if and only if for anyx,y∈S(E) withP(Bxy) > ,x+_y<  onBxy[], Theorem ..

element inL_{(F,K). For an arbitrarily chosen representativeξ}_{ofξ, define twoF-random}

variables (ξ₎–_and|ξ_{|respectively by}

ξ–(ω) = ⎧ ⎨ ⎩



ξ_(ω) ξ(ω)= ,

 otherwise,

and

ξ_{(ω) =}ξ_{(ω), ∀ω∈.}

Then the equivalence class of (ξ)–, denoted byξ–, is calledthe generalized inverseof ξ; the equivalence class of|ξ|, denoted by|ξ|, is calledthe absolute valueofξ. When ξ∈L_{(F,C), setξ=u+iv, whereu,v∈L}_(F,R),ξ¯_{:=u–ivis calledthe complex conjugate}

ofξ andsgn(ξ) :=|ξ|–_{·ξ is calledthe signofξ. It is obvious that|ξ|=|¯ξ|,ξ·sgn(ξ) =}¯ _|ξ|,

|sgn(ξ)|=IAandξ–·ξ=ξ·ξ–=IA, whereA= [ξ= ].

3 Random smoothness

In this section, (E, · ) is always an RN module overKwith base (,F,P) and (E∗, · ∗) its random conjugate space.

Proposition . Let f be an element in E∗.Thenf∗=∨{Ref(x) :x∈S(E)}andf∗= ∨{Ref(x) :x∈U(E)}.

Proof It is known in [], Proposition ., thatf∗=∨{|f(x)|:x∈U(E)}and in [],

Proposition ., thatf∗=∨{|f(x)|:x∈S(E)}. For anyx∈S(E), takex=sgn(f(x))x, then

x∈S(E) andf(x) =sgn(f(x))f(x) =|f(x)|, so that|f(x)|=Ref(x)≤ ∨{Ref(x) :x∈S(E)}, which justifies the equality f∗ =∨{Ref(x) :x∈S(E)}. It is not difficult to see that

f∗=∨{Ref(x) :x∈U(E)}.

The notion of support functionals is a preparation for that of random smoothness.

Definition . LetAbe a subset ofE, then a nonzero elementf inE∗is called a support functional forAif there exists somexinAsuch thatRef(x) =∨{Ref(x) :x∈A}, in which

casexis called a support point ofA, the setH(f,x) ={x∈E:Ref(x) =Ref(x)}is called

a support hyperplane for A, and the support functionalf and the support hyperplane

H(f,x) are both said to supportAatx.

Remark . By Theorem . each pointxinS(E) is a support point ofU(E) and therefore

gives rise to at least one support hyperplane forU(E) that supportsU(E) atx.

Proposition . For any f,f∈S(E∗)with P(Aff) > and x∈S(E)such that both fand

fsupport U(E)at x,namely,Refj(x) =∨{Refj(x) :x∈U(E)},j= ,  (this time,Refj(x) =

fj∗=IA_fj,j= , ),the following statements hold:

() LetD∈Fbe such thatD⊂Aff andP(D) > .IfH(IDf,x) =H(IDf,x),then

H(IGf,x) =H(IGf,x)for anyG∈FwithG⊂DandP(G) > ;

() For an arbitraryD∈FwithD⊂Aff andP(D) > ,H(IDf,x) =H(IDf,x)if and

Proof (). Letx∈H(IGf,x), then

ReIDf(IGx+ID\Gx) =ReIDf(IGx) +ReIDf(ID\Gx)

=ReIGf(x) +ReID\Gf(x)

=ReIGf(x) +ReID\Gf(x)

=IDRef(x) =ReIDf(x),

so thatIGx+ID\Gx∈H(IDf,x) =H(IDf,x), which implies

ReIDf(IGx+ID\Gx) =ReIGf(x) +ReID\Gf(x) =ReIDf(x),

thusReIGf(x) =ReIGf(x), from which we can see thatH(IGf,x)⊂H(IGf,x).

Simi-larly,H(IGf,x)⊂H(IGf,x).

(). (⇐) is clear.

(⇒). We only need to show thatIDRef=IDRef. Assume by way of contradiction that

there existsxinEsuch thatP(D) > , whereD= [IDRef(x)=IDRef(x)]∩D. Let

D_= IDRef

x= ∩ IDRef

x= ∩D,

D_= IDRef

x= ∩ IDRef

x= ∩D,

D_= IDRef

x= ∩ IDRef

x= ∩ IDRef

x=IDRef

x∩D,

thenD=D_∪D_ ∪D_. It is enough to show thatP(D) = . IfP(D_) > , letx_=ID_(Ref(x))–x, then

Ref

x_=ID_

Ref

x–Ref

x=ID_=ID_IAf_=ID_Ref(x),

which implies thatx_∈H(ID_f,x). SinceH(IDf,x) =H(IDf,x) andD⊂D, by () of

this proposition we have x ∈H(IDf,x), namely, ReIDf(x

) =ReIDf(x) =ID. On

the other hand,ReID_f(x) =ID_Ref(x) =ID_(Ref(x))–Ref(x) = . The contradiction

shows thatP(D_) = . Similarly,P(D_) = .

Now let us show thatP(D_) =  as follows. IfP(D_) > , then lety=I_D

(Ref(x

₎₎–_{x. It is easy to see thatRef}

(y) =ID(Ref(x

₎₎–_×

Ref(x) =ID_, which implies that ReID_f(y) =ID_ =ID_IAf_ =ReID_f(x), namely,y∈ H(ID_f,x). Again by () of this proposition we have H(ID_f,x) =H(ID_f,x).

Con-sequently, ReI_D

f(y) =ReIDf(x) =ID. On the other hand, by the definition of y

and D_ we have that ReID_f(y) =ReID_f(ID_(Ref(x))–x) =ReID_(Ref(x))–f(x) = ID_(Ref(x))–Ref(x)=  onD, which contradicts the fact that

ReID_f(y) =ReID_f(x) =ID_.

ThusP(D_) = .

Definition . In an RN module (E,·), an elementxofS(E) is called a point of random

smoothness ofU(E) ifH(IAfff,x) =H(IAfff,x) for any two support functionalsf,f

forU(E) atxwithP(Aff) > .Eis said to be random smooth if each point ofS(E) is a

point of random smoothness ofU(E).

Proposition . Suppose that E is an RN module and that x∈S(E).Then the following

are equivalent.

() xis a point of random smoothness ofU(E);

() If bothfandfinS(E∗)supportU(E)atxandP(Aff) > ,thenIAfff=IAfff;

() IffandfinS(E∗)satisfyReIAfffj(x) =IAff(j= , ),thenIAfff=IAfff;

() IffandfinS(E∗)satisfyf(x) =f(x) =IAff,thenIAfff=IAfff.

Proof ()⇒(). Assume thatIAfff=IAfff, thenIAfff–f

∗_{= . It is easy to see that}

P(D) > , whereD=Aff∩[f–f∗= ]. ConsideringIDf,IDf∈S(E∗) andIDx∈S(E),

we can gteReIDfj(x) =IDRefj(x) =ID· ∨{Refj(x) :x∈U(E)}=∨{ReIDfj(x) :x∈U(E)} (j= , ). Namely, both IDf andIDf are support functionals forU(E) atx. By

Defini-tion . we haveH(IDf,x) =H(IDf,x), and then by Proposition .()IDf=IDf, which

leads toIDf–f∗= , a contradiction.

()⇒(). Observing thatReIAfffj(x) =IAff =IAfffj

∗_=∨{ReI

Afffj(x) :x∈U(E)}(j=

, ), namely, bothIAf_f_f andIAf_f_f supportU(E) atx, we obtain thatIAf_f_f=IAf_f_f

by ().

()⇒() is obvious.

()⇒() Suppose thatf,f∈E∗are two support functionals forU(E) withP(Aff) > ,

namely,

Refj(x) =∨

Refj(x) :x∈U(E)

=fj∗ (j= , ).

It is enough to show thatH(IAfff,x) =H(IAfff,x) by Proposition .(). Clearly,

ReIAf_f_

f∗

–

f(x) =ReIAf_f_

f∗

–

f(x) =IAf_f_,

IAff

f∗

– f,IAff

f∗

–

f∈S

E∗.

From

f∗=Ref(x)≤f(x)≤ f∗x=f∗·IAx_≤ f∗

it follows that|f(x)|=f∗, namely,|Ref(x) –iRef(ix)|=f∗, which together with Ref(x) =f∗ implies thatRef(ix) = ; consequently,f(x) =Ref(x) =f∗. In the

same way,f(x) =f∗. ThusIAff(f

∗₎–_f

(x) =IAff(f

∗₎–_f

(x) =IAff. Then by ()

IAff(f

∗₎–_f

=IAff(f

∗₎–_f

. Therefore,H(IAfff,x) =H(IAfff,x).

Corollary . Suppose that E is an RN module and that x∈S(E).Then the following are

equivalent.

() Eis random smooth;

() For eachx∈S(E),if bothfandfinS(E∗)supportU(E)atxwithP(Aff) > ,then

() For eachx∈S(E),iffandfinS(E∗)satisfyReIAfff(x) =ReIAfff(x) =IAff,then

IAf_f_f=IAf_f_f;

() For eachx∈S(E),iffandfinS(E∗)satisfyIAf_f_f(x) =IAf_f_f(x) =IAf_f_,then IAf_f_f=IAf_f_f.

The relations between random smoothness and random strict convexity are described by Theorems . and . below.

Theorem . Suppose that E is an RN module.If E∗is random strictly convex,then E is

random smooth.

Proof Suppose thatEis not random smooth, by Corollary ., there exist x∈S(E) and

f,f∈S(E∗) such that

IAf_f_f(x) =IAf_f_f(x) =IAf_f_ ()

and

IAfff=IAfff. ()

From () it follows that



IAff(f+f)(x) =IAff,

which yields

_IAff(f+f)

=IAff,

namely,

IAf_f_f+IAf_f_f= IAf_f_=IAf_f_f+IAf_f_f.

By the random strict convexity ofE∗, there existsξ ∈L

+ such that ξ >  onAff and

IAf_f_f=ξIAf_f_f, so thatIAf_f_f=ξIAf_f_f, which impliesξIAf_f_ =IAf_f_, thusIAf_f_f=

IAf_f_f, a contradiction to ().

Theorem . Suppose that E is an RN module.If E∗is random smooth,then E is random

strictly convex.

Proof Assume by way of contradiction thatEis not random strictly convex, then there

existx,x∈S(E) andD∈FwithD⊂BxxandP(D) >  such that

ID x+x

 =ID.

By Theorem ., there existsf∈E∗such that

f

ID

x+x



=ID x+x

andf∗=ID. Noticing that

ID=

ID f(x) +

ID f(x)≤

ID

f(x)+

ID

f(x)≤ID, ()

we have

ID

f(x)+

ID

f(x)=ID,

where ID

|f(x)| ≤ 

ID and ID|f(x)| ≤ 

ID. Consequently,ID|f(xi)|=ID(i= , ). By ()

ID=ID_ Ref(x) +I_DRef(x), which in combination withID|f(x)|=ID|f(x)|=IDyields

Ref(x) =Ref(x) =ID, and henceIDf(x) =IDf(x) =ID. Using the canonical embed-ding mapping (see Remark .), we haveJ(IDx)(f) =f(IDx) =f(IDx) =J(IDx)(f) =ID,

thusJ(IDxj)∗∗=∨{J(IDxj)(f) :f ∈U(E∗)}=ID=J(IDxj)(f) (j= , ), namely, bothJ(IDx)

andJ(IDx) supportU(E∗) atf. SinceE∗is random smooth, by Corollary . we can

ob-tain thatIDJ(IDx) =IDJ(IDx) so thatIDJ(IDx–IDx)∗∗= , which yieldsIDx–x= ,

a contradiction.

Proposition . Let E be a random reflexive RN module.Then the following hold.

() Eis random strictly convex if and only ifE∗is random smooth; () Eis random smooth if and only ifE∗is random strictly convex.

Proof Since the canonical embedding mappingJis random-norm preserving and linear,

andJ(E) =E∗∗, it is a straightforward verification by Theorems . and ..

4 Gâteaux differentiability of random norm

In classical normed spaces the functiont→(x+ty–x)/tfromR\{}toRplays an

important role in Gâteaux differentiability of the classical norm []. Motivated by this, we consider in an RN module (E, · ) the mappingf defined by

f(t,y) =x+ty–x

t , ∀t∈L

_{(F,R)\{},∀y∈E,}

where xis a fixed element inS(E). It is feasible to define the Gâteaux differentiability

of random norm viaf(t,y) and establish its relation to random smoothness. It should be pointed out that some terminologies and properties in this section are closely related to [], Section .

For anyD∈FwithP(D) >  andt,tinL(F,R) such that  <t<tonD, we can verify

that

ID

x+ty–x

t

≤ID

x+ty–x

t

,

ID

x–ty–x

–t ≤

ID

x–ty–x

–t

and

ID

x–ty–x

–t ≤

ID

x+ty–x

as in the classical case, see [] for details. Consequently,f(t,y) is nondecreasing intin the sense thatIDf(t,y)≤IDf(t,y) for anyt,t∈L(F,R) andD∈FwithP(D) >  such

thatt,t=  onDandt<tonD. Besides,f(t,y) possesses the following properties as

described in Lemma ..

Lemmas .() and . below are implied by [], Lemma ., and [], Theorem .(), respectively. Here we present slightly different proofs to illustrate the typical stratification analysis on random normed modules and to keep self-contained.

Lemma . ()∧{IDf(t,y) :t∈L(F,R),t> on D}=ID· ∧{f(s,y) :s∈R,s> }; ()∧{IDf(ξs,y) :s∈R,s> }=ID· ∧{f(s,y) :s∈R,s> },∀ξ∈L(F,R),ξ> on D; ()∨{IDf(t,y) :t∈L(F,R),t< on D}=ID· ∨{f(s,y) :s∈R,s< }.

Proof (). It is clear that

∧IDf(t,y) :t∈L(F,R),t>  onD

≤ID· ∧

f(s,y) :s∈R,s> .

For any fixedt∈L_{(F,R) witht>  onD, denoteD}

(t) =D∩[t≥] andDn(t) =D∩[_n_+≤

t<_n],∀n≥, thenD=∞_n=Dn(t). Since

IDn(t)f(t,y)≥IDn(t)f



n+ ,y

≥IDn(t)· ∧

f(s,y) :s∈R,s> , ∀n,

we obtain that

IDf(t,y) =

∞

n=

IDn(t)f(t,y)≥

∞

n=

IDn(t)· ∧

f(s,y) :s∈R,s> 

=ID· ∧

f(s,y) :s∈R,s> .

The proofs of () and () are similar.

Lemma . Denote G+(x,y) =∧{f(t,y) :t∈R,t> },G–(x,y) =∨{f(t,y) :t∈R,t< }.

Then G+(x,y)and G–(x,y)satisfy the following:

() G+(x,ξy) =ξG+(x,y),G–(x,ξy) =ξG–(x,y);

() G+(x,y+y)≤G+(x,y) +G+(x,y),

where x∈S(E),y,y,y∈E,ξ∈L+.

Proof (). For anyξ∈L

+, letA= [ξ> ], then by Lemma .

G+(x,ξy) =∧

f(t,ξy) :t∈R,t> 

=∧

x+tξy–x

t :t∈R,t> 

=∧

IAx+tξy–IAx

t :t∈R,t> 

=IAξ· ∧

IA·

x+tξy–x

tξ :t∈R,t> 

=IAξ· ∧

IA·

x+ty–x

t :t∈R,t> 

=IAξG+(x,y)

=ξG+(x,y).

The proof of the other equality is similar. (). Notice that

∧f(t,y) :t∈R,t> 

+∧f(t,y) :t∈R,t> 

=∧f(t,y) +f(t,y) :t∈R,t> 

. ()

In fact, for any positive numberstandt,

f(t,y) +f(t,y)≥ f(t,y) +f(t,y)

∧ f(t,y) +f(t,y)

≥ ∧f(t,y) +f(t,y) :t∈R,t> 

.

Consequently, the left-hand side of () is not less than the right-hand side, so that () can be easily verified. Then, by the following

f(t,y+y) =

x+t(y+y)–x

t

≤x+ ty–x

t +

x+ ty–x

t

=f(t,y) +f(t,y) (∀t∈R,t> )

it is easy to see that

∧f(t,y+y) :t∈R,t> 

≤ ∧f(t,y) +f(t,y) :t∈R,t> 

=∧f(t,y) :t∈R,t> 

+∧f(t,y) :t∈R,t> 

,

which implies that

G+(x,y+y)≤G+(x,y) +G+(x,y).

The Gâteaux differentiability was defined for proper local functions from a random lo-cally convex module toL¯_{(F) in [], Definition .. Since a random locally convex}

mod-ule is a generalization of a random normed modmod-ule and a random norm isL_{-convex, it}

is easy to see that a random norm is also a proper local function. For random norms we present the following definition of Gâteaux differentiability, which is equivalent to that in [], Definition .. Under Definition . we can establish the relations among supporting functionals, points of random smoothness and Gâteaux differentiability of random norms.

Definition . Let (E, · ) be an RN module,x∈S(E) andy∈E. ThenG–(x,y) and G+(x,y) are respectively called the left-hand and right-hand Gâteaux derivative of the

the directionyifG–(x,y) =G+(x,y), in which case the common value ofG–(x,y) and G+(x,y) is denoted byG(x,y) and is called the Gâteaux derivative of · atxin the

directiony. If · is Gâteaux differentiable atxin every directiony∈E, then it is called

Gâteaux differentiable atx. Finally, if · is Gâteaux differentiable at every point of the

random unit sphereS(E), then · is said to be Gâteaux differentiable.

Remark . It is obvious thatG–(x,y)≤G+(x,y),G–(x, –y) = –G+(x,y) and G–(x,x) = G+(x,x) =G(x,x) =x=IAx for anyxinS(E) andyinE.

Lemma . Let(E, · )be an RN module,x∈S(E)and f ∈S(E∗)such that P(D) > ,

where D=Ax∩Af.Then IDf supports U(E)at xif and only if

IDG–(x,y)≤IDRef(y)≤IDG+(x,y), ∀y∈E. ()

Proof (⇒). By the following facts

ReIDf(x) =∨

ReIDf(x) :x∈U(E)

=IDf∗=ID,

IDf(x) =ReIDf(x) –iReIDf(ix) and IDf(x)≤IDf∗x=ID,

one can obtain thatIDf(x) =ID. Since

IDRef(ty) =IDRef(x+ty) –IDRef(x)≤ID

x+ty–x

for any positive numberstand anyy∈E, the following inequalities hold:

ID

x–ty–x

–t = –ID

x+t(–y)–x t

≤–ID

Ref(t(–y))

t =IDRef(y)≤ID

x+ty–x

t ,

which implies ().

(⇐). SinceG–(x,x) =x=G+(x,x), it follows thatIDRef(x) =IDby (). Noticing that

ReIDf(x) =ID=∨

ReIDf(x) :x∈U(E)

,

we complete the proof.

Theorem . Let(E, · )be an RN module and xbe a point in S(E).Then the following

statements hold:

() xis a point of random smoothness ofU(E)if and only if · is Gâteaux

differentiable atx;

() Ifxis a point of random smoothness ofU(E)andf inS(E∗)supportsU(E)atx,

thenIAfG(x,y) =ReIAff(y).

Proof (). (⇒). Assume by way of contradiction that · is not Gâteaux differentiable

where M= [G–(x,y) <G+(x,y)]. SinceIAcxG–(x,y) =IAcxG+(x,y) =IAcxyand

IAcy_G–(x,y) =IAcy_G+(x,y) = , one knows thatM⊂Axand thatM⊂Ay.

Take an arbitrarysinL_{(F,R) such that}

G–(x,y) <s<G+(x,y) onM. ()

DefineW={ry:r∈L(F,R)}andfs(ry) =IMrs,∀r∈L(F,R), then it is clear thatW is anL_{(F,R)-submodule ofE, thatf}

sis anL-linear function onW, and that

fs(ry) =IMrs≤IMrG+(x,y) =IMG+(x,ry), ∀r∈L+.

For anyr∈L(F,R)\L₊, denoteD=M∩[r< ], thenfs(–IDry) = –IDrs≥–IDrG–(x,y) = G–(x, –IDry) = –G+(x,IDry), and hencefs(IDry)≤G+(x,IDry) =IDG+(x,ry).

Com-bining the fact that fs(IM\Dry) =IM\Drs ≤IM\DG+(x,ry), we can see that fs(ry)≤ IDG+(x,ry) +IM\DG+(x,ry) =IMG+(x,ry). Thus

fs(ry)≤IMG+(x,ry), ∀r∈L(F,R).

By Theorem . there exists anL-linear functionx∗_s:E→L(F,R) such thatx∗_s|W=fs andx∗_s(y)≤G+(x,y),∀y∈E.

Without loss of generality, suppose thatEis a complex RN module. Let

gs(y) =x∗_s(y) –ix∗_s(iy).

It is easy to show thatgs:E→L(F,C) isL-linear andRegs=x∗s, from which it follows thatRegs(y)≤G+(x,y),∀y∈E. Then we obtain

Regs(y)≤G+(x,y)≤

x+y–x

 ≤ y, ∀y∈E

and

Regs(y) = –Regs(–y)≥–G+(x, –y) =G–(x,y),

which together with the fact that gs∗ =Regs∗ yields that gs ∈E∗, gs∗ ≤ and

G–(x,x)≤Regs(x)≤G+(x,x). ThusRegs(x) =IAx and soIAxgs

∗_=I

Ax. Namely,

IAx_gssupportsU(E) atx.

Therefore, for anysandssatisfying () andsIM=sIM, there exist two corresponding

gs andgs inE∗ such that bothIAxgs andIAxgs supportU(E) atxandIAxgs∗=

IAxgs∗=IAx. Noticing that

IAxRegs(y) =IAxx

∗

s(y) =IAxfs(y) =IMs

IAx_Regs(y) =IAx_x∗s(y) =IAx_fs(y) =IMs,

(⇐). Suppose that · is Gâteaux differentiable atx,f inS(E∗) supportsU(E) atx.

Then it is obvious thatAf ⊂Ax. By Lemma . we have

IAf Ref(y) =IAfG(x,y), ∀y∈E. ()

IffandfinS(E∗) both supportU(E) atxandP(Aff) > , thenIAffRef=IAffRef,

which implies thatIAfff=IAfff. By Proposition .xis a point of random smoothness

ofU(E).

(). It is clear by ().

Remark . It should be pointed out that further development of random smoothness under Definition . confronted some obstacles, one of which is how to establish the con-nection between random smoothness of an RN moduleEand classical smoothness of the abstractLpspace derived fromE. It is an interesting problem and attracts some attention in different literature [, ].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented in this manuscript has been accomplished through contributions of both authors. Both authors have read and approved the final manuscript.

Author details

1_{School of Mathematics and Statistics, Chongqing Technology and Business University, Xuefu Road No. 21, Chongqing,}

400067, P.R. China.2_{School of Mathematics and Statistics, Central South University, Xiaoxiang Middle Road No. 405,}

Changsha, 410083, P.R. China.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11301568, 11271388), Science Foundation of Chongqing Municipal Education Commission (Grant No. KJ120732), and Science Foundation of Chongqing Technology and Business University (2012-56-10).

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 12 February 2017 Accepted: 28 April 2017

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