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 SLip₁(X)${\mathrm{SLip}}_1(X)$ 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 SLip₁(Y)${\mathrm{SLip}}_1(Y)$ onto SLip₁(X)${\mathrm{SLip}}_1(X)$ can be written in the form Tf=c⋅(f∘τ)+ϕ$Tf=c\cdot (f\circ \tau )+\varphi$, where τ   is an isometry, c>0$c>0$ and ϕ∈SLip₁(X)$\varphi \in {\mathrm{SLip}}_1(X)$. 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 SLip₁(X)${\mathrm{SLip}}_1(X)$ and SLip₁(Y)${\mathrm{SLip}}_1(Y)$.