Let X and Y be two pointed metric spaces.In this article,we give a generalization of the Cheng-Dong-Zhang theorem for coarse Lipschitz embeddings as follows:If f:X→Y is a standard coarse Lipschitz embedding,then for ...Let X and Y be two pointed metric spaces.In this article,we give a generalization of the Cheng-Dong-Zhang theorem for coarse Lipschitz embeddings as follows:If f:X→Y is a standard coarse Lipschitz embedding,then for each x^(*)∈Lip_(0)(X)there existα,γ>0 depending only on f and Q_(x)*∈Lip_(0)(Y)with‖Q_(x)*‖_(Lip)≤α‖x^(*)‖_(Lip)such that|Q_(x)*f(x)-x^(*)(x)|≤γ‖x^(*)‖_(Lip),for all x∈X.Coarse stability for a pair of metric spaces is studied.This can be considered as a coarse version of Qian Problem.As an application,we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual.Indeed,we show that X is not a Lipschitz retract of its bidual if X is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract.If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space,then the problem also has a negative answer for a separable space.展开更多
Suppose that X, Y are two real Banach Spaces. We know that for a standard ε-isometry f : X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of *. In this paper, by ...Suppose that X, Y are two real Banach Spaces. We know that for a standard ε-isometry f : X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of *. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ε-isometry to be stable in assuming that N is w*-closed in Y*.Making use of this result, we improve several known results including Figiel’s theorem in reflexive spaces.We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f)≡span[f(X)] contains a complemented linear isometric copy of X;Moreover, if X =Y, then for every e-isometry f: X → X, there exists a surjective linear isometry S:X → X such that f-S is uniformly bounded by 2ε on X.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.12126329,12171266,12126346,12101234)Simons Foundation(Grant No.585081)+6 种基金Educational Commission of Fujian Province(Grant No.JAT190589)Natural Science Foundation of Fujian Province(Grant No.2021J05237)the research start-up fund of Jimei University(Grant No.ZQ2021017)the research start-up fund of Putian University(Grant No.2020002)the Natural Science Foundation of Hebei Province(Grant No.A2022502010)the Fundamental Research Funds for the Central Universities(Grant No.2023MS164)the Natural Science Foundation of Fujian Province(Grant No.2023J01805)。
文摘Let X and Y be two pointed metric spaces.In this article,we give a generalization of the Cheng-Dong-Zhang theorem for coarse Lipschitz embeddings as follows:If f:X→Y is a standard coarse Lipschitz embedding,then for each x^(*)∈Lip_(0)(X)there existα,γ>0 depending only on f and Q_(x)*∈Lip_(0)(Y)with‖Q_(x)*‖_(Lip)≤α‖x^(*)‖_(Lip)such that|Q_(x)*f(x)-x^(*)(x)|≤γ‖x^(*)‖_(Lip),for all x∈X.Coarse stability for a pair of metric spaces is studied.This can be considered as a coarse version of Qian Problem.As an application,we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual.Indeed,we show that X is not a Lipschitz retract of its bidual if X is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract.If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space,then the problem also has a negative answer for a separable space.
基金supported in part by the Natural Science Foundation of China(Grant Nos.11731010,11471270&11471271)
文摘Suppose that X, Y are two real Banach Spaces. We know that for a standard ε-isometry f : X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of *. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ε-isometry to be stable in assuming that N is w*-closed in Y*.Making use of this result, we improve several known results including Figiel’s theorem in reflexive spaces.We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f)≡span[f(X)] contains a complemented linear isometric copy of X;Moreover, if X =Y, then for every e-isometry f: X → X, there exists a surjective linear isometry S:X → X such that f-S is uniformly bounded by 2ε on X.
