This is such a article to consider an ＂into＂ isometric mapping between two unit spheres of two infinite dimensional spaces of different types. In this article, we find a useful condition （using the Krein-Milman pro...This is such a article to consider an ＂into＂ isometric mapping between two unit spheres of two infinite dimensional spaces of different types. In this article, we find a useful condition （using the Krein-Milman property） under which an into-isometric mapping from the unit sphere of e（Γ） to the unit sphere of a normed space E can be linearly isometric extended.展开更多
In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of Lp （μ, H） and Lp（v, g）（p 〉 2, H is a Hilbert space）, and -Vo（ S（ np（μ, H） ） ） Vo（ S（ Lp（μ, H） ） ） then Vo can ...In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of Lp （μ, H） and Lp（v, g）（p 〉 2, H is a Hilbert space）, and -Vo（ S（ np（μ, H） ） ） Vo（ S（ Lp（μ, H） ） ） then Vo can be extended to a linear isometry defined on the whole space. If 1 〈 p 〈 2 and Vo is an ＂anti-l-Lipschitz＂ mapping, then Vo can also be linearly and isometrically extended.展开更多
In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.
In this paper,we first derive the representation theorem of onto isometric mappings in theunit spheres of l~1(F) type spaces,and then conclude that such mappings can be extended to the wholespace as real linear isomet...In this paper,we first derive the representation theorem of onto isometric mappings in theunit spheres of l~1(F) type spaces,and then conclude that such mappings can be extended to the wholespace as real linear isometrics by using a previous result of the author.展开更多
This paper considers the isometric extension problem concerning the mapping from the unitsphere S(E)of the normed space E into the unit sphere S(l~∞(Γ)).We find a condition under whichan isometry from S,(E)into S1(l...This paper considers the isometric extension problem concerning the mapping from the unitsphere S(E)of the normed space E into the unit sphere S(l~∞(Γ)).We find a condition under whichan isometry from S,(E)into S1(l~∞(Γ))can be linearly and isometrically extended to the whole space.Since l~∞(Γ)is universal with respect to isometry for normed spaces,isometric extension problemson a class of normed spaces are solved.More precisely,if E and F are two normed spaces,and ifV:S(E)→S(F)is a surjective isometry,where c(Γ)■(Γ),then Vcan be extended tobe an isometric operator defined on the whole space.展开更多
In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of two ALP-spaces with p 〉 2 and -Vo（S1（LP）） C Vo（S1（LP））, then V0 can be extended to a linear isometry defined on the whole spa...In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of two ALP-spaces with p 〉 2 and -Vo（S1（LP）） C Vo（S1（LP））, then V0 can be extended to a linear isometry defined on the whole space. If 1 〈 p 〈 2 and Vo is an ＂anti-l-Lipschitz＂ mapping, then Vo can also be linearly and isometrically extended.展开更多
This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from th...This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.展开更多
In this article,we use some analytic and geometric characters of the smooth points in a sphere to study the isometric extension problem in the separable or reflexive real Banach spaces.We obtain that under some condit...In this article,we use some analytic and geometric characters of the smooth points in a sphere to study the isometric extension problem in the separable or reflexive real Banach spaces.We obtain that under some condition the answer to this problem is affirmative.展开更多
In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces lp（Γ） and lp（△）（p 〉1）. We first derive the representation of isometries between the unit spheres of comple...In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces lp（Γ） and lp（△）（p 〉1）. We first derive the representation of isometries between the unit spheres of complex Banach spaces lp（Γ） and lp（△）. Then we arrive at a conclusion that any surjective isometry between the unit spheres of complex Banach spaces lp（Γ）and lp（△） can be extended to be a linear isometry on the whole space.展开更多
In this article, we prove that an into 1-Lipschitz mapping from the unit sphere of a Hilbert space to the unit sphere of an arbitrary normed space, which under some conditions, can be extended to be a linear isometry ...In this article, we prove that an into 1-Lipschitz mapping from the unit sphere of a Hilbert space to the unit sphere of an arbitrary normed space, which under some conditions, can be extended to be a linear isometry on the whole space.展开更多
In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space s（n）. Moreover, the author presents the representation for the onto isometric map...In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space s（n）. Moreover, the author presents the representation for the onto isometric mappings in the space s. It is obtained that if V is a surjective isometry from the space s onto s with V（0） = 0, then V must be real linear.展开更多
Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some ...Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.展开更多
Rate of force development(RFD)and impulse(IMP)are important mechanical measures of muscular performance but are relatively unused within the rehabilitation and performance community.Due principally to access to low-co...Rate of force development(RFD)and impulse(IMP)are important mechanical measures of muscular performance but are relatively unused within the rehabilitation and performance community.Due principally to access to low-cost testing devices and understanding the utility of these measures.The aim of this study therefore was to quantify the reliability of various force-time variables using load-cell technology collected via isometric knee extension whilst constrained in an isokinetic device(CON90)or unconstrained on a physiotherapy plinth at 60 and 90 degree angles(UNCON60 and UNCON90).Thirty-two volunteers had their peak force(PF),RFD,peak RFD(PRFD),and IMP assessed across three protocols.For all variables,UNCON60 had the largest variability across all measures.PF and PRFD were found to have small variability(ICC>0.67 and CV<10%).With regards to RFD 2080 all three protocols were found to have moderate variability all ICCs above 0.75,however,all CVs were greater than 10%ranging from~11%-22%.Finally,IMP 2080 was found to have moderate variability for both CON90 and UNCON90,the absolute consistency once more greater than 10%(~11%-25%).Using the constrained and unconstrained protocols,PF and PRFD can be measured reliably between trials with 90 degree knee position.展开更多
In this paper, we study the extension of isometries between the unit spheres of normed space E and C（Ω）. We obtain that any surjective isometry between the unit spheres of normed space E and C（Ω） can be extended...In this paper, we study the extension of isometries between the unit spheres of normed space E and C（Ω）. We obtain that any surjective isometry between the unit spheres of normed space E and C（Ω） can be extended to be a linear isometry on the whole space E and give an affirmative answer to the corresponding Tingley＇s problem （where Ω be a compact metric space）.展开更多
In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p（μ） （1 〈 p 〈∞, p ≠ 2） and a Banach space E can be extended to a linear isometry fr...In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p（μ） （1 〈 p 〈∞, p ≠ 2） and a Banach space E can be extended to a linear isometry from L^p（μ） onto E, which means that if the unit sphere of E is （metrically） isometric to the unit sphere of L^P（μ）, then E is linearly isometric to L^p（μ）. We also prove that every surjective 1-Lipschitz or anti-l-Lipschitz map between the unit spheres of L^p（μ1, H1） and L^p（μ2, H2） must be an isometry and can be extended to a linear isometry from L^p（μ2, H2） to L^p（μ2, H2）, where H1 and H2 are Hilbert spaces.展开更多
In this paper, we shall present a short and simple proof on the isometric linear extension problem of into-isometries between two unit spheres of atomic abstract L^p-spaces （0 〈 p 〈 ∞）.
In this paper, we study the extension of isometries between the unit spheres of normed space E and lP（p 〉 1）. We arrive at a conclusion that any surjective isometry between the unit sphere of normed space lP（p 〉 ...In this paper, we study the extension of isometries between the unit spheres of normed space E and lP（p 〉 1）. We arrive at a conclusion that any surjective isometry between the unit sphere of normed space lP（p 〉 1） and E can be extended to be a linear isometry on the whole space lP（p 〉 1） under some condition.展开更多
Let T be a mapping from the unit sphere S[l^p（Г）] into S[l^p（△）] of two atomic AL^p- spaces. We prove that if T is a 1-Lipschitz mapping such that -T[S[l^p（Г）]] belong to T[S[l^p（Г）]], then T can be linear...Let T be a mapping from the unit sphere S[l^p（Г）] into S[l^p（△）] of two atomic AL^p- spaces. We prove that if T is a 1-Lipschitz mapping such that -T[S[l^p（Г）]] belong to T[S[l^p（Г）]], then T can be linearly isometrically extended to the whole space for p 〉 2; if T is injective and the inverse mapping T^-1 is a 1-Lipschitz mapping, then T can be extended to be a linear isometry from l^p（Г） into l^p（△） for 1 〈 p ≤ 2.展开更多
In this paper, we prove that an into isometry form S（l（n）^∞） to S（E）, which under some conditions, can be extended to be a linear isometry defined on the whole space. Therefore we improve the results of [Ding, ...In this paper, we prove that an into isometry form S（l（n）^∞） to S（E）, which under some conditions, can be extended to be a linear isometry defined on the whole space. Therefore we improve the results of [Ding, G. G.： The isometric extension of an into mapping from the unit sphere S（l（2）^∞） to S（Lμ^1）. Acta Mathematica Sinica, English Series, 22（6）, 1721-1724 （2006）].展开更多
The main result of this paper is to prove Fang and Wang＇s result by another method： Let E be any normed linear space and Vo ： S（E）→ S（l^1） be a surjective isometry. Then V0 can be linearly isometrically extend...The main result of this paper is to prove Fang and Wang＇s result by another method： Let E be any normed linear space and Vo ： S（E）→ S（l^1） be a surjective isometry. Then V0 can be linearly isometrically extended to E.展开更多
基金supported by the National Natural Science Foundation of China(10871101)the Research Fund for the Doctoral Program of Higher Education (20060055010)
文摘This is such a article to consider an ＂into＂ isometric mapping between two unit spheres of two infinite dimensional spaces of different types. In this article, we find a useful condition （using the Krein-Milman property） under which an into-isometric mapping from the unit sphere of e（Γ） to the unit sphere of a normed space E can be linearly isometric extended.
基金Supported by the National Natural Science Foundation of China(Grant No.11271248)Specific Academic Discipline Project of Shanghai Municipal Education Commission(Grant No.11xk11)
文摘In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of Lp （μ, H） and Lp（v, g）（p 〉 2, H is a Hilbert space）, and -Vo（ S（ np（μ, H） ） ） Vo（ S（ Lp（μ, H） ） ） then Vo can be extended to a linear isometry defined on the whole space. If 1 〈 p 〈 2 and Vo is an ＂anti-l-Lipschitz＂ mapping, then Vo can also be linearly and isometrically extended.
基金supported by Research Foundation for Doctor Programme (Grant No. 20060055010)National Natural Science Foundation of China (Grant No. 10871101)
文摘In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.
基金supported by National Science Foundation of China(19971046)the Doctoral Programme Foundation of Ministry of Education of China
文摘In this paper,we first derive the representation theorem of onto isometric mappings in theunit spheres of l~1(F) type spaces,and then conclude that such mappings can be extended to the wholespace as real linear isometrics by using a previous result of the author.
基金Natural Science Foundation of Guangdong Province,China (Grant No.7300614)
文摘This paper considers the isometric extension problem concerning the mapping from the unitsphere S(E)of the normed space E into the unit sphere S(l~∞(Γ)).We find a condition under whichan isometry from S,(E)into S1(l~∞(Γ))can be linearly and isometrically extended to the whole space.Since l~∞(Γ)is universal with respect to isometry for normed spaces,isometric extension problemson a class of normed spaces are solved.More precisely,if E and F are two normed spaces,and ifV:S(E)→S(F)is a surjective isometry,where c(Γ)■(Γ),then Vcan be extended tobe an isometric operator defined on the whole space.
基金Supported by National Natural Science Foundation of China (Grant No. 10871101)Research Fund for the Doctoral Program of Higher Education (Grant No. 20060055010)
文摘In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of two ALP-spaces with p 〉 2 and -Vo（S1（LP）） C Vo（S1（LP））, then V0 can be extended to a linear isometry defined on the whole space. If 1 〈 p 〈 2 and Vo is an ＂anti-l-Lipschitz＂ mapping, then Vo can also be linearly and isometrically extended.
基金This paper is supported by The National Natural Science Foundation of China(10571090)The Research Fund for the Doctoral Program of Higher Education(20010055013)
文摘This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.
基金Supported by National Natural Science Foundation of China(Grant No.11371201)
文摘In this article,we use some analytic and geometric characters of the smooth points in a sphere to study the isometric extension problem in the separable or reflexive real Banach spaces.We obtain that under some condition the answer to this problem is affirmative.
基金supported by Higher Educational Science and Technology Program Foundation of Shandong Province(J11LA02)Young and Middle-Aged Scientists Research Foundation of Shandong Province(BS2010SF004)Higher Educational Science and Technology Program Foundation of Shandong Province(J10LA53)
文摘In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces lp（Γ） and lp（△）（p 〉1）. We first derive the representation of isometries between the unit spheres of complex Banach spaces lp（Γ） and lp（△）. Then we arrive at a conclusion that any surjective isometry between the unit spheres of complex Banach spaces lp（Γ）and lp（△） can be extended to be a linear isometry on the whole space.
基金Supported by NSFC (10871101)the Doctoral Programme Foundation of Institution of Higher Education (20060055010)
文摘In this article, we prove that an into 1-Lipschitz mapping from the unit sphere of a Hilbert space to the unit sphere of an arbitrary normed space, which under some conditions, can be extended to be a linear isometry on the whole space.
基金Supported by the National Science Foundation of China (10571090) The Research Fund for the Doctoral Program of Higher Education (20010055013)
文摘In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space s（n）. Moreover, the author presents the representation for the onto isometric mappings in the space s. It is obtained that if V is a surjective isometry from the space s onto s with V（0） = 0, then V must be real linear.
基金supported by National Natural Science Foundation of China (Grant No.10871101)the Research Fund for the Doctoral Program of Higher Education (Grant No. 20060055010)
文摘Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.
文摘Rate of force development(RFD)and impulse(IMP)are important mechanical measures of muscular performance but are relatively unused within the rehabilitation and performance community.Due principally to access to low-cost testing devices and understanding the utility of these measures.The aim of this study therefore was to quantify the reliability of various force-time variables using load-cell technology collected via isometric knee extension whilst constrained in an isokinetic device(CON90)or unconstrained on a physiotherapy plinth at 60 and 90 degree angles(UNCON60 and UNCON90).Thirty-two volunteers had their peak force(PF),RFD,peak RFD(PRFD),and IMP assessed across three protocols.For all variables,UNCON60 had the largest variability across all measures.PF and PRFD were found to have small variability(ICC>0.67 and CV<10%).With regards to RFD 2080 all three protocols were found to have moderate variability all ICCs above 0.75,however,all CVs were greater than 10%ranging from~11%-22%.Finally,IMP 2080 was found to have moderate variability for both CON90 and UNCON90,the absolute consistency once more greater than 10%(~11%-25%).Using the constrained and unconstrained protocols,PF and PRFD can be measured reliably between trials with 90 degree knee position.
文摘In this paper, we study the extension of isometries between the unit spheres of normed space E and C（Ω）. We obtain that any surjective isometry between the unit spheres of normed space E and C（Ω） can be extended to be a linear isometry on the whole space E and give an affirmative answer to the corresponding Tingley＇s problem （where Ω be a compact metric space）.
基金Supported by the Fundamental Research Funds for the Central UniversitiesNational Natural Science Foundation of China (Grant No. 10871101)
文摘In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p（μ） （1 〈 p 〈∞, p ≠ 2） and a Banach space E can be extended to a linear isometry from L^p（μ） onto E, which means that if the unit sphere of E is （metrically） isometric to the unit sphere of L^P（μ）, then E is linearly isometric to L^p（μ）. We also prove that every surjective 1-Lipschitz or anti-l-Lipschitz map between the unit spheres of L^p（μ1, H1） and L^p（μ2, H2） must be an isometry and can be extended to a linear isometry from L^p（μ2, H2） to L^p（μ2, H2）, where H1 and H2 are Hilbert spaces.
基金This work is supported by The National Natural Science Foundation of China （Grant No. 10501090） and The Research Fund for the Doctoral Program of Higher Education （No. 20010055013）
文摘In this paper, we shall present a short and simple proof on the isometric linear extension problem of into-isometries between two unit spheres of atomic abstract L^p-spaces （0 〈 p 〈 ∞）.
基金Supported by Natural Science Foundation of China (Grant No. 10571090)The second author is supported by NSFC (Grant No. 10571090)the Doctoral Program Foundation of Institution of Higher Education (Grant No. 20060055010)
文摘In this paper, we study the extension of isometries between the unit spheres of normed space E and lP（p 〉 1）. We arrive at a conclusion that any surjective isometry between the unit sphere of normed space lP（p 〉 1） and E can be extended to be a linear isometry on the whole space lP（p 〉 1） under some condition.
基金the Natural Science Foundation of the Education Department of Jiangsu Province (No.06KJD110092)
文摘Let T be a mapping from the unit sphere S[l^p（Г）] into S[l^p（△）] of two atomic AL^p- spaces. We prove that if T is a 1-Lipschitz mapping such that -T[S[l^p（Г）]] belong to T[S[l^p（Г）]], then T can be linearly isometrically extended to the whole space for p 〉 2; if T is injective and the inverse mapping T^-1 is a 1-Lipschitz mapping, then T can be extended to be a linear isometry from l^p（Г） into l^p（△） for 1 〈 p ≤ 2.
基金Supported by National Natural Science Foundation of China (Grant No. 10871101)the Doctoral Pr0grame Foundation of Institution of Higher Education (Grant No. 20060055010)
文摘In this paper, we prove that an into isometry form S（l（n）^∞） to S（E）, which under some conditions, can be extended to be a linear isometry defined on the whole space. Therefore we improve the results of [Ding, G. G.： The isometric extension of an into mapping from the unit sphere S（l（2）^∞） to S（Lμ^1）. Acta Mathematica Sinica, English Series, 22（6）, 1721-1724 （2006）].
基金Foundation item： the National Natural Science Foundation of China （No. 10571090） the Research Fund for the Doctoral Program of Higher Education （No. 20060055010） and the Fund of Tianjin Educational Comittee （No. 20060402）.
文摘The main result of this paper is to prove Fang and Wang＇s result by another method： Let E be any normed linear space and Vo ： S（E）→ S（l^1） be a surjective isometry. Then V0 can be linearly isometrically extended to E.
