Near isometries of Bochner \(L^ 1\) and \(L^{\infty}\) spaces (Q1081079)

It is a result of Benyamini that, as a consequence of known properties of spaces of scalar-valued continuous functions, if two spaces \(L^ p(\mu_ 1)\) and \(L^ p(\mu_ 2)\) are nearly isometric, for either \(p=1\) or \(p=\infty\), then they are isometric. Here we show that the same conclusion can be drawn for near isometries of certain Bochner spaces. Let \((\Omega_ i,\Sigma_ i,\mu_ i)\) be \(\sigma\)-finite measure spaces for \(i-=1,2\), and let E be a Hilbert space. We prove that if the Bochner spaces \(L^ p(\Omega_ 1,.\Sigma_ 1,\mu_ 1,E)\) and \(L^ p(\Omega_ 2,\Sigma_ 2,\mu_ 2,E)\) are nearly isometric, for either \(p=1\) or \(p=\infty\), then \(L^ 1(\Omega_ 1,\Sigma_ 1,\mu_ 1,E)\) is isometric to \(L^ 1(\Omega_ 2,\Sigma_ 2,\mu_ 2,E)\) and hence \(L^{\infty}(\Omega_ 1,\Sigma_ 1,\mu_ 1,E)\) is isometric to \(L^{\infty}(\Omega_ 2,\Sigma_ 2,\mu_ 2,E)\).