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- W197661642 abstract "We prove that a Banach space $X$ has the metric approximation property if and only if $mathcal F(Y,X)$ is an ideal in $mathcal L(Y,X^{**})$ for all Banach spaces $Y$. Furthermore, $X^*$ has the metric approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $phi : X^* rightarrow X^{***}$ there exists a Hahn-Banach extension operator $Phi : {mathcal F(Y,X)}^* rightarrow {mathcal L(Y,X^{**})}^*$ such that $Phi(x^* otimes y^{**}) = (phi x^*) otimes y^{**}$ for all $x^* in X^*$ and all $y^{**} in Y^{**}$. We also prove that $X^*$ has the approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $phi : X^* rightarrow X^{***}$ there exists a Hahn-Banach extension operator $Phi : {mathcal F(Y,X)}^* rightarrow {mathcal W(Y,X^{**})}^*$ such that $Phi(x^* otimes y^{**}) = (phi x^*) otimes y^{**}$ for all $x^* in X^*$ and all $y^{**} in Y^{**}$, which in turn is equivalent to $mathcal F(Y,hat{X})$ being an ideal in $mathcal W(Y,hat{X}^{**})$ for all Banach spaces $Y$ and all equivalent renormings $hat{X}$ of $X$." @default.
- W197661642 created "2016-06-24" @default.
- W197661642 creator A5073481459 @default.
- W197661642 date "2003-12-01" @default.
- W197661642 modified "2023-10-04" @default.
- W197661642 title "Approximation properties for dual spaces" @default.
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- W197661642 doi "https://doi.org/10.7146/math.scand.a-14425" @default.
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