A generalization of the parallelogram equality in normed spaces
A normed space \(X\) is called quasi-inner product space, if for \(x,y\in X\), \[ \|x+y\|^4-\|x-y\|^4= 8\|x\|^2 g(x,y)+ 8\|y\|^2 g(y,x), \] holds, where \[ g(x,y)= {\|x\|\over 2} \Biggl[\lim_{t\to 0_-} {\|x+ty\|-\|x\|\over t}+ \lim_{t\to 0_+} {\|x+ ty\|-\|x\|\over t}\Biggr]. \] The author proves: A quasi-inner product space is very smooth.
scientific article; zbMATH DE number 1295042