Some results on random smoothness

Theorem 2.1

[4]

Let E be a left module over the algebra \(L^{0}({\mathcal {F}},R)\), \(M\subset E\) be an \(L^{0}({\mathcal {F}},R)\)-submodule, \(f:M\rightarrow L^{0}({\mathcal {F}},R)\) be an \(L^{0}\)-linear function and \(p:E\rightarrow L^{0}({\mathcal {F}},R)\) be an \(L^{0}\)-sublinear function such that \(f(x)\leq p(x),\forall x\in M\). Then there exists an \(L^{0}\)-linear function \(g:E\rightarrow L^{0}({\mathcal {F}},R)\) such that g extends f and \(g(x)\leq p(x),\forall x\in E\).

Definition 2.2

[20]

Definition 2.3

[20]

Let \((E, \Vert \cdot \Vert )\) be an RN module. A linear operator f from E to \(L^{0}({\mathcal {F}}, K)\) is an almost surely (briefly, a.s.) bounded random linear functional on E if there exists some ξ in \(L^{0}_{+}\) such that \(\vert f(x) \vert \leq\xi\cdot \Vert x \Vert , \forall x\in E\). Denote by \(E^{\ast}\) the linear space of all a.s. bounded random linear functionals on E with the ordinary pointwise addition and scalar multiplication on linear operators. Then \((E^{\ast}, \Vert \cdot \Vert ^{\ast})\) is an RN module over K with base \((\Omega,{\mathcal {F}},P)\), in which the module multiplication \(\cdot:L^{0}({\mathcal {F}},K)\times E^{\ast}\rightarrow E^{\ast}\) is defined by \((\xi\cdot f)(x)=\xi\cdot(f(x)), \forall\xi\in L^{0}({\mathcal {F}}, K), f\in E^{\ast}\) and \(x\in E\), and the random norm \(\Vert \cdot \Vert ^{\ast}: E^{\ast}\rightarrow L^{0}_{+}\) is defined by \(\Vert f \Vert ^{\ast} =\wedge\{ \xi\in L^{0}_{+}: \vert f(x) \vert \leq\xi\cdot \Vert x \Vert , \forall x\in E\}, \forall f\in E^{\ast}\). \((E^{\ast}, \Vert \cdot \Vert ^{\ast})\) is called the random conjugate space of \((E, \Vert \cdot \Vert )\).

Remark 2.4

Theorem 2.5

[4, 22]

Let \((E, \Vert \cdot \Vert )\) be an RN module over K with base \((\Omega,{\mathcal {F}},P)\), \(M\subset E\) be a linear subspace, and \(f:M\rightarrow L^{0}({\mathcal {F}}, K)\) be an a.s. bounded random linear functional on M. Then there exists \(F\in E^{\ast}\) such that (1) \(F(x)=f(x),\forall x\in M\), and (2) \(\Vert F \Vert ^{\ast}= \Vert f \Vert ^{\ast}\). As a consequence, for any \(x\in E\), there exists \(g\in E^{\ast}\) such that \(g(x)= \Vert x \Vert \) and \(\Vert g \Vert ^{\ast}=I_{A_{x}}\).

Definition 2.6

[15]

For any \(x, y\) in an RN module \((E, \Vert \cdot \Vert )\) over K with base \((\Omega,{\mathcal {F}},P)\), \([ \Vert x \Vert \neq0]\) is denoted by \(A_{x}\), called the support of x, and we briefly write \(A_{xy}=A_{x}\cap A_{y}\) and \(B_{xy}=A_{x}\cap A_{y}\cap A_{x-y}\). The random unit sphere of E is defined by \(S(E)=\{x\in E: \Vert x \Vert =\tilde{I}_{A}\text{ for some } A\in{\mathcal {F}} \text{ with } P(A)>0\}\). The random closed unit ball of E is defined by \(U(E)=\{x\in E: \Vert x \Vert \leq1\}\).

Definition 2.7

[15]

An RN module \((E, \Vert \cdot \Vert )\) is said to be random strictly convex if for any x and \(y\in E\backslash\{\theta\}\) such that \(\Vert x+y \Vert = \Vert x \Vert + \Vert y \Vert \) there exists \(\xi\in L^{0}_{+}\) such that \(\xi>0\) on \(A_{xy}\) and \(I_{A_{xy}}x=\xi(I_{A_{xy}}y)\).

Proposition 3.1

Let f be an element in \(E^{\ast}\). Then \(\Vert f \Vert ^{*}= \vee\{ \operatorname{Re}f(x):x\in S(E)\}\) and \(\Vert f \Vert ^{*}= \vee\{\operatorname{Re}f(x):x\in U(E)\}\).

Proof

It is known in [21], Proposition 2.2, that \(\Vert f \Vert ^{\ast}=\vee \{ \vert f(x) \vert :x\in U(E)\}\) and in [15], Proposition 2.1, that \(\Vert f \Vert ^{*}= \vee \{ \vert f(x) \vert :x\in S(E)\}\). For any \(x\in S(E)\), take \(x^{\prime }=\operatorname{sgn}(\overline{f(x)})x\), then \(x^{\prime}\in S(E)\) and \(f(x^{\prime })=\operatorname{sgn}(\overline{f(x)})f(x)= \vert f(x) \vert \), so that \(\vert f(x) \vert =\operatorname{Re}f(x^{\prime })\leq\vee\{\operatorname{Re}f(x):x\in S(E)\}\), which justifies the equality \(\Vert f \Vert ^{*}= \vee\{\operatorname{Re}f(x):x\in S(E)\}\). It is not difficult to see that \(\Vert f \Vert ^{*}= \vee\{\operatorname{Re}f(x):x\in U(E)\}\). □

Definition 3.2

Let A be a subset of E, then a nonzero element f in \(E^{\ast}\) is called a support functional for A if there exists some \(x_{0}\) in A such that \(\operatorname{Re}f(x_{0})=\vee\{\operatorname{Re}f(x):x\in A\}\), in which case \(x_{0}\) is called a support point of A, the set \(H(f,x_{0})=\{x\in E:\operatorname{Re}f(x)=\operatorname{Re}f(x_{0})\}\) is called a support hyperplane for A, and the support functional f and the support hyperplane \(H(f,x_{0})\) are both said to support A at \(x_{0}\).

Remark 3.3

By Theorem 2.5 each point \(x_{0}\) in \(S(E)\) is a support point of \(U(E)\) and therefore gives rise to at least one support hyperplane for \(U(E)\) that supports \(U(E)\) at \(x_{0}\).

Proposition 3.4

Proof

(2). (⇐) is clear.

Similarly, \(P(D_{2}^{\prime})=0\).

Now let us show that \(P(D_{3}^{\prime})=0\) as follows.

Therefore, \(P(D^{\prime})=0\) and this completes the proof. □

Definition 3.5

In an RN module \((E, \Vert \cdot \Vert )\), an element \(x_{0}\) of \(S(E)\) is called a point of random smoothness of \(U(E)\) if \(H(I_{A_{f_{1}f_{2}}}f_{1},x_{0})=H(I_{A_{f_{1}f_{2}}}f_{2},x_{0})\) for any two support functionals \(f_{1},f_{2}\) for \(U(E)\) at \(x_{0}\) with \(P(A_{f_{1}f_{2}})>0\). E is said to be random smooth if each point of \(S(E)\) is a point of random smoothness of \(U(E)\).

Proposition 3.6

Proof

(1)⇒(2). Assume that \(I_{A_{f_{1}f_{2}}}f_{1}\neq I_{A_{f_{1}f_{2}}}f_{2}\), then \(I_{A_{f_{1}f_{2}}} \Vert f_{1}-f_{2} \Vert ^{\ast}\neq0\). It is easy to see that \(P(D)>0\), where \(D=A_{f_{1}f_{2}}\cap[ \Vert f_{1}-f_{2} \Vert ^{\ast}\neq0]\). Considering \(I_{D}f_{1},I_{D}f_{2}\in S(E^{\ast})\) and \(I_{D}x_{0}\in S(E)\), we can gte \(\operatorname{Re}I_{D}f_{j}(x_{0})=I_{D}\operatorname{Re}f_{j}(x_{0})=I_{D}\cdot\vee\{\operatorname{Re}f_{j}(x):x\in U(E)\}=\vee \{\operatorname{Re}I_{D}f_{j}(x):x\in U(E)\}\) (\(j=1,2\)). Namely, both \(I_{D}f_{1}\) and \(I_{D}f_{2}\) are support functionals for \(U(E)\) at \(x_{0}\). By Definition 3.5 we have \(H(I_{D}f_{1},x_{0})=H(I_{D}f_{2},x_{0})\), and then by Proposition 3.4(2) \(I_{D}f_{1}=I_{D}f_{2}\), which leads to \(I_{D} \Vert f_{1}-f_{2} \Vert ^{\ast}=0\), a contradiction.

(2)⇒(3). Observing that \(\operatorname{Re}I_{A_{f_{1}f_{2}}}f_{j}(x_{0})=I_{A_{f_{1}f_{2}}}= \Vert I_{A_{f_{1}f_{2}}}f_{j} \Vert ^{\ast }=\vee\{\operatorname{Re}I_{A_{f_{1}f_{2}}}f_{j}(x):x\in U(E)\} (j=1,2)\), namely, both \(I_{A_{f_{1}f_{2}}}f_{1}\) and \(I_{A_{f_{1}f_{2}}}f_{2}\) support \(U(E)\) at \(x_{0}\), we obtain that \(I_{A_{f_{1}f_{2}}}f_{1}=I_{A_{f_{1}f_{2}}}f_{2}\) by (2).

(3)⇒(4) is obvious.

Corollary 3.7

Theorem 3.8

Suppose that E is an RN module. If \(E^{\ast}\) is random strictly convex, then E is random smooth.

Proof

Theorem 3.9

Suppose that E is an RN module. If \(E^{\ast}\) is random smooth, then E is random strictly convex.

Proof

Proposition 3.10

Proof

Since the canonical embedding mapping J is random-norm preserving and linear, and \(J(E)=E^{\ast\ast}\), it is a straightforward verification by Theorems 3.8 and 3.9. □

Lemma 4.1

(1) \(\wedge\{I_{D}f(t,y):t\in L^{0}({\mathcal {F}},R),t>0\textit{ on }D\}=I_{D}\cdot\wedge\{f(s,y):s\in R,s>0\}\);

(2) \(\wedge\{I_{D}f(\xi s,y):s\in R,s>0\}=I_{D}\cdot\wedge\{ f(s,y):s\in R,s>0\},\forall\xi\in L^{0}({\mathcal {F}},R),\xi>0\) on D;

(3) \(\vee\{I_{D}f(t,y):t\in L^{0}({\mathcal {F}},R),t<0\textit{ on }D\}=I_{D}\cdot\vee\{f(s,y):s\in R,s<0\}\).

Proof

The proofs of (2) and (3) are similar. □

Lemma 4.2

Proof

Definition 4.3

Let \((E, \Vert \cdot \Vert )\) be an RN module, \(x_{0}\in S(E)\) and \(y\in E\). Then \(G_{-}(x_{0},y)\) and \(G_{+}(x_{0},y)\) are respectively called the left-hand and right-hand Gâteaux derivative of the random norm \(\Vert \cdot \Vert \) at \(x_{0}\) in the direction y. \(\Vert \cdot \Vert \) is called Gâteaux differentiable at \(x_{0}\) in the direction y if \(G_{-}(x_{0},y)=G_{+}(x_{0},y)\), in which case the common value of \(G_{-}(x_{0},y)\) and \(G_{+}(x_{0},y)\) is denoted by \(G(x_{0},y)\) and is called the Gâteaux derivative of \(\Vert \cdot \Vert \) at \(x_{0}\) in the direction y. If \(\Vert \cdot \Vert \) is Gâteaux differentiable at \(x_{0}\) in every direction \(y\in E\), then it is called Gâteaux differentiable at \(x_{0}\). Finally, if \(\Vert \cdot \Vert \) is Gâteaux differentiable at every point of the random unit sphere \(S(E)\), then \(\Vert \cdot \Vert \) is said to be Gâteaux differentiable.

Remark 4.4

It is obvious that \(G_{-}(x,y)\leq G_{+}(x,y), G_{-}(x,-y)=-G_{+}(x,y)\) and \(G_{-}(x,x)=G_{+}(x,x)=G(x,x)= \Vert x \Vert =I_{A_{x}}\) for any x in \(S(E)\) and y in E.

Lemma 4.5

Proof

Theorem 4.6

Proof

(1). (⇒). Assume by way of contradiction that \(\Vert \cdot \Vert \) is not Gâteaux differentiable at \(x_{0}\). Then there exists \(y_{0}\) in E such that \(G_{-}(x_{0},y_{0})< G_{+}(x_{0},y_{0})\), namely, \(P(M)>0\), where \(M=[G_{-}(x_{0},y_{0})< G_{+}(x_{0},y_{0})]\). Since \(I_{A_{x_{0}}^{c}}G_{-}(x_{0},y_{0})=I_{A_{x_{0}}^{c}}G_{+}(x_{0},y_{0})=I_{A_{x_{0}}^{c}} \Vert y_{0} \Vert \) and \(I_{A_{y_{0}}^{c}}G_{-}(x_{0},y_{0})=I_{A_{y_{0}}^{c}}G_{+}(x_{0},y_{0})=0\), one knows that \(M\subset A_{x_{0}}\) and that \(M\subset A_{y_{0}}\).

By Theorem 2.1 there exists an \(L^{0}\)-linear function \(x_{s}^{\ast }:E\rightarrow L^{0}({\mathcal {F}}, R)\) such that \(x_{s}^{\ast}| _{W}=f_{s}\) and \(x_{s}^{\ast}(y)\leq G_{+}(x_{0},y),\forall y\in E\).

(2). It is clear by (7). □

Remark 4.7

It should be pointed out that further development of random smoothness under Definition 3.5 confronted some obstacles, one of which is how to establish the connection between random smoothness of an RN module E and classical smoothness of the abstract \(L^{p}\) space derived from E. It is an interesting problem and attracts some attention in different literature [19, 24].