A reverse of Bessel's inequality in 2-inner product spaces and some Grüss type related results with applications (Q815997)

Let \(\left( X;\left( .,.| .\right) \right) \) be a 2-inner product space over the real or complex number field \(\mathbb{K}\) and \(\left\{ e_{i}\right\} _{i\in I}\) be a family of linearly independent vectors in \(X\) which is \(z\)-orthonormal for some \(z\in X,\) that is \(\left( e_{i},e_{j}| z\right) =\delta _{ij},\) \(i,j\in I,\) where \(\delta _{ij}\) is the Kronecker delta. If for a finite part \(F\) of \(I\) there are the numbers \(\varphi _{i},\Phi _{i}\in \mathbb{ K}\) \(\left( i\in F\right) \) such that \(\text{Re}(\sum_{i\in F}\Phi _{i}e_{i}-x,x-\sum_{i\in F}\varphi _{i}e_{i}| z) \geq 0,\) the authors prove the following reverse of Bessel's inequality \[ 0\leq \left( x,x| z\right) -\sum_{i\in F}\left| \left( x,e_{i}| z\right) \right| ^{2}\leq \tfrac{1}{4}\sum_{i\in F}\left| \Phi _{i}-\varphi _{i}\right| ^{2}. \]