Inner product spaces and quadratic functional equations

In this paper, we introduce the functional equations f(2x−y)+f(x+2y)=5[f(x)+f(y)],f(2x−y)+f(x+2y)=5f(x)+4f(y)+f(−y),f(2x−y)+f(x+2y)=5f(x)+f(2y)+f(−y),f(2x−y)+f(x+2y)=4[f(x)+f(y)]+[f(−x)+f(−y)].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} f(2x-y)+f(x+2y)&=5\bigl[f(x)+f(y)\bigr], \\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\ f(2x-y)+f(x+2y)&=4\bigl[f(x)+f(y)\bigr]+\bigl[f(-x)+f(-y)\bigr]. \end{aligned}$$ \end{document} We show that these functional equations are quadratic and apply them to characterization of inner product spaces. We also investigate the stability problem on restricted domains. These results are applied to study the asymptotic behaviors of these quadratic functions in complete β-normed spaces.