Equations which characterize inner product spaces

Abstract:

It is shown that if $N$ is a normed linear space and there is a point $y$ of norm 1 such that an inequality of the type ${a^2}||x|{|^2} \leqq {\lim _{u \to 0}}G(\{ ||{b_i}ux + {c_i}y||\} _{i = 1}^n) \leqq {b^2}||x|{|^2}$ holds for all $x$ in $N$ (where $0 < a \leqq b$, the ${c_i}$’s are nonzero and $G$ and $|| \cdot ||$ satisfy a certain twice-differentiability condition), then $N$ is isomorphic to an inner product space and $\inf ||T|| \cdot ||{T^{ - 1}}|| \leqq b/a$, where the infimum is taken over all linear homeomorphisms $T$ between $N$ and an inner product space. In the event that $a = b = 1$, the inequality reduces to an equation which characterizes inner product spaces. An example shows that these results do not follow without the twice-differentiability condition on $G$.