Operator norm attainment and inner product spaces

In this paper we prove that a finite dimensional real normed linear space X$\mathbb{X}$ is an inner product space iff for any linear operator T   on X$\mathbb{X}$, T   attains its norm at _e1,_e2∈_SX$e_1,e_2\in S_{\mathbb{X}}$ implies T   attains its norm at span{_e1,_e2}∩_SX$\mathit{span}\{e_1,e_2\}\cap S_{\mathbb{X}}$.