Grüss type inequalities in normed spaces (Q432329)

Generalizations of the classical Grüss inequality in normed spaces are given. Let \((V,\|\cdot \|)\) be a normed vector space over \(\mathbb{F} \, (\mathbb{F}=\mathbb{R} \,\text{or} \, \mathbb{C})\) with \(V^*\) the space of all linear continuous functionals on \(V\). The acting of a functional \(v^* \in V^*\) on \(x\in V\) is denoted by \((x,v^*)\). The symbol \(\| \cdot \|_*\) stands for the norm \(\| v^*\|_*=\sup_{0\not= x\in V} \frac{|(x,v^*)|}{\| x\|}\). We assume that \(V\) is equipped with two preorders \(\prec_1\) and \(\prec_2\). For given \(x,y\in V\) and \(v^*, z^* \in V^*\) with \((x,v^*)\not= 0\) the Chebyshev-type functional is defined as NEWLINE\[NEWLINE T_{x,v^*} (y,z^*)=(y,z^*)(x,v^*) -(x,z^*)(y,v^*).NEWLINE\]NEWLINENEWLINENEWLINETheorem. Let \(\prec_k\) (\(k=1,2\)) be translation-invariant preorders on \(V\) such that there exists \(N\) such that \(-v\prec_1 z\prec_2 v\) implies \(\| z\| \leq N \|v\|\) for any \(z,v\in V\). Assume that \(x\in V\), \(v^*\in V^*\) with \((x,v^*)\not= 0 \) and let \(I=[a,b]_{\prec_{1,2}} \subset V\) be a proper vectorial interval.NEWLINENEWLINEIf \(y\in I\) and \(a+b\in \mathrm{span}\{x\}\), then for any \(z^* \in V^*\) NEWLINE\[NEWLINE |T_{x,v^*} (y,z^*)| \leq \frac{N}{2} |(x,v^*)| \;\| b-a\|\;\|z^*-\lambda v^*\|_{*}, NEWLINE\]NEWLINE where \(\lambda =(x,z^*)/(x,v^*)\).NEWLINENEWLINEAlso, variants of the previous theorem in normed algebras and in lattice-ordered algebras are given. The results are illustrated with some specific examples.