The quasi-representation theorem of the conjugate cone \([L^{\beta} (\mu , X)]_{\beta}^*\;(0 < \beta < 1)\) (Q1044351)

Let \((\Omega, \mathcal{M}, \mu)\) be a regular Borel finite measure space, \((X, \| \cdot \|)\) a Banach space, and \(L^\beta (\mu,X)\) (\(0<\beta<1\)) the \(\beta\)-Banach space of Bochner integrable functions equipped with the usual \(\beta\)-norm \(\|f\|_\beta= \int_\Omega \|f\|^\beta \;d \mu(t)\). The paper under review is concerned with the representation problem of the conjugate cone \([L^\beta(\mu,X)]_\beta^*\) of \(L^\beta (\mu,X)\). The author proves that \[ [L^\beta(\mu,X)]_\beta^* \approxeq L^\infty M^+ (\mu,S). \]