A weak stability bound for the symmetrization Θ = (f(·) − f(−·))/2 of a general ε-isometry f from a Banach space X to a Banach space Y is presented. As a corollary, the following somewhat surprising weak stability result is obtained: For every x* ∈ X*, there exists ϕ ∈ Y* with ‖ϕ‖ = ‖x*‖ ≔ r such that \(\mid\langle{x}^*,x\rangle-\langle\varphi,\Theta(x)\rangle\mid\;\leqslant\frac{3}{2}r\varepsilon\;\;\;{\rm{for\;all}}\;x\in{X}.\)This result is used to prove new stability theorems for the symmetrization Θ of f.

中文翻译：

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Banach空间上ε-对称的对称性

提出了从Banach空间X到Banach空间Y的一般ε-等轴测图f的对称化Θ=（f（·）-f（-·））/ 2的弱稳定性。作为推论，获得以下有些令人惊讶弱稳定性结果：对于每一个X *∈ X *，存在φ ∈ ý *与‖ φ ‖=‖ X *‖≔ ř使得 \（\中间\ langle {X} ^ *，x \ rangle- \ langle \ varphi，\ Theta（x）\ rangle \ mid \; \ leqslant \ frac {3} {2} r \ varepsilon \; \; \; {\ rm {for \; all }} \; x \ in {X}。\）该结果用于证明f的对称性θ的新稳定性定理。