A functional representation of almost isometries

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• 作者：Javier Cabello^a ; ^{coco@unex.es" class="auth_mail" title="E-mail the corresponding author} ; Jesú ; s A. Jaramillo^b ; ^{jaramil@mat.ucm.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Almost isometries ; Quasi-metric spaces ; Semi-Lipschitz functions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：445
• 期：2
• 页码：1243-1257
• 全文大小：388 K

文摘

For each quasi-metric space X   we consider the convex lattice class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300646&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300646&_rdoc=1&_issn=0022247X&md5=73d170bd093a4c7962301eabb2fd286c" title="Click to view the MathML source">SLip₁(X)class="mathContainer hidden">class="mathCode">${\mathrm{SLip}}_1(X)$ of all semi-Lipschitz functions on X with semi-Lipschitz constant not greater than 1. If X and Y are two complete quasi-metric spaces, we prove that every convex lattice isomorphism T   from class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300646&_mathId=si2.gif&_user=111111111&_pii=S0022247X16300646&_rdoc=1&_issn=0022247X&md5=b4deaabf136e8b9beb506b64e120a194" title="Click to view the MathML source">SLip₁(Y)class="mathContainer hidden">class="mathCode">${\mathrm{SLip}}_1(Y)$ onto class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300646&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300646&_rdoc=1&_issn=0022247X&md5=73d170bd093a4c7962301eabb2fd286c" title="Click to view the MathML source">SLip₁(X)class="mathContainer hidden">class="mathCode">${\mathrm{SLip}}_1(X)$ can be written in the form class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300646&_mathId=si3.gif&_user=111111111&_pii=S0022247X16300646&_rdoc=1&_issn=0022247X&md5=54425078f26769b5dbd71ab2c823ee92" title="Click to view the MathML source">Tf=c⋅(f∘τ)+ϕclass="mathContainer hidden">class="mathCode">$Tf=c\cdot (f\circ \tau )+\varphi$, where τ   is an isometry, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300646&_mathId=si4.gif&_user=111111111&_pii=S0022247X16300646&_rdoc=1&_issn=0022247X&md5=822340d2f50262e7d3d96db1a5cd6792" title="Click to view the MathML source">c>0class="mathContainer hidden">class="mathCode">$c>0$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300646&_mathId=si5.gif&_user=111111111&_pii=S0022247X16300646&_rdoc=1&_issn=0022247X&md5=78cdbc654e668a0bddd45855bddce9fa" title="Click to view the MathML source">ϕ∈SLip₁(X)class="mathContainer hidden">class="mathCode">$\varphi \in {\mathrm{SLip}}_1(X)$. As a consequence, we obtain that two complete quasi-metric spaces are almost isometric if, and only if, there exists an almost-unital convex lattice isomorphism between class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300646&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300646&_rdoc=1&_issn=0022247X&md5=73d170bd093a4c7962301eabb2fd286c" title="Click to view the MathML source">SLip₁(X)class="mathContainer hidden">class="mathCode">${\mathrm{SLip}}_1(X)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300646&_mathId=si2.gif&_user=111111111&_pii=S0022247X16300646&_rdoc=1&_issn=0022247X&md5=b4deaabf136e8b9beb506b64e120a194" title="Click to view the MathML source">SLip₁(Y)class="mathContainer hidden">class="mathCode">${\mathrm{SLip}}_1(Y)$.