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Norm attaining bilinear forms on \(L_1(\mu)\). (English) Zbl 0964.46008

Given a real or complex Banach space \(X\) and a natural number \(N\), let \({\mathcal L}^N(X)\) denote the space of all continuous \(N\)-linear forms on \(X\) and we say that \(\phi\in{\mathcal L}^N(X)\) attain its norm if there are \(x_1,x_2,\dots, x_N\in B_X\) (the closed unit ball of \(X\)) such that \[ |\phi(x_1,x_2,\dots, x_N)|= \|\phi\|:= \sup\{|\phi(y_1,\dots, y_N)|: y_1,\dots y_N\in B_X\}. \] Let \({\mathcal A}{\mathcal L}^N(X)\) be the set of norm attaining continuous \(N\)-linear forms on \(X\). The question is whether \({\mathcal A}{\mathcal L}^N(X)\) is dense in \({\mathcal L}^N(X)\) or not. It is shown that given a finite measure \(\mu\), the set of norm attaining bilinear forms is dense in the space of all continuous bilinear forms on \(L_1(\mu)\) if and only if \(\mu\) is purely atomic.
Main result: Given a finite measure \(\mu\), the following statements are equivalent:
(1) \(\mu\) is purely atomic.
(2) \({\mathcal A}{\mathcal L}^N(L_1(\mu))\) is dense in \({\mathcal L}^N(L_1(\mu))\) for any natural number \(N\).
(3) \({\mathcal A}{\mathcal L}^N(L_1(\mu))\) is dense in \({\mathcal L}^N(L_1(\mu))\) for some \(N\geq 2\).
(4) \({\mathcal A}{\mathcal L}^2(L_1(\mu))\) is dense in \({\mathcal L}^2(L_1(\mu))\).

MSC:

46B28 Spaces of operators; tensor products; approximation properties
46B20 Geometry and structure of normed linear spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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