On Universally Left-stability of ε-Isometry
Let X, Y be two real Banach spaces and ε≥0. A map f : X → Y is said to be a standard ε-isometry if│││f/(x) - f(y)││ - ]ix - Y││x-y││ ε for all x,y C X and with f(O) = O. We say that a pair of Banach spaces (X, Y) is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometr...
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Published in | 数学学报:英文版 no. 11; pp. 2037 - 2046 |
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Main Author | |
Format | Journal Article |
Language | English |
Published |
01.11.2013
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Subjects | |
Online Access | Get full text |
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Summary: | Let X, Y be two real Banach spaces and ε≥0. A map f : X → Y is said to be a standard ε-isometry if│││f/(x) - f(y)││ - ]ix - Y││x-y││ ε for all x,y C X and with f(O) = O. We say that a pair of Banach spaces (X, Y) is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometry f : X → Y, there is a bounded linear operator T : L(f) → f(X) → X so that ││Tf(x) - x││ ≤γε for all x E X. X(Y) is said to be universally left-stable if (X, Y) is always stable for every Y(X). In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w*-closed subspace of ∞ (1) for some set F, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant. |
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Bibliography: | Let X, Y be two real Banach spaces and ε≥0. A map f : X → Y is said to be a standard ε-isometry if│││f/(x) - f(y)││ - ]ix - Y││x-y││ ε for all x,y C X and with f(O) = O. We say that a pair of Banach spaces (X, Y) is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometry f : X → Y, there is a bounded linear operator T : L(f) → f(X) → X so that ││Tf(x) - x││ ≤γε for all x E X. X(Y) is said to be universally left-stable if (X, Y) is always stable for every Y(X). In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w*-closed subspace of ∞ (1) for some set F, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant. ε-Isometry, linear isometry, stability, injective space, Banach space 11-2039/O1 |
ISSN: | 1439-8516 1439-7617 |
DOI: | 10.1007/s10114-013-2585-2 |