Grothendieck type inequalities
One version of Grothendieck's inequality states that for compact spaces \(K_1\) and \(K_2\) and any bounded bilinear form \(\Phi: C(K_1) \times C(K_2) \rightarrow {\mathbb R}\), there exist probability measures \(\mu_1\) and \(\mu_2\) on \(K_1\) and \(K_2\), respectively, such that \[ |\Phi(f,g)|\leq K_G^{\mathbb R}\|\Phi\|\bigg(\int_{K_1} f^2\,d\mu_1\bigg)^{1/2} \bigg(\int_{K_2} g^2\,d\mu_2\bigg)^{1/2}, \] for all \(f\in C(K_1)\) and \(g\in C(K_2)\), where \(\pi/2 \leq K_G^{\mathbb R} \leq \pi/(2 \ln (1+\sqrt{2}))\) denotes the (real) Grothendieck constant. The author of the present paper presents analogues of the above result within the setting of pseudocompact spaces for the case of \textit{positive} bilinear forms. In particular, it follows that the above inequality holds for positive bilinear forms with constant \(1\) instead of \(K_G^{\mathbb R}\). The proofs rely on a representation theorem for bilinear forms from [\textit{W.\,Adamski}, Glas.\ Mat., III.\ Ser.\ 26, No.\,1--2, 31--44 (1991; Zbl 0808.28008)] and the (reverse) Hölder inequality.
scientific article; zbMATH DE number 5669321