A real vector space is an inner product space if every two dimensional subspace is an inner product space ?

Question

Is it true that a vector space over the field of real numbers is an inner product space if every two dimensional subspace is an inner product space ? does it have anything to do with Neuman-Jordan's result that if in a real normed linear space the parallelogram equality $||x+y||^2+||x-y||^2=2(||x||^2+||y|^2)$ holds always , then the norm comes from an inner product space ?

Answer 1

I'm going to assume that by inner product space you mean that there is an inner product $\langle -,-\rangle$ that gives the specific norm. In that case under the given conditions we have that in each two dimensional subspace there is such a form satisfying $$\|x+y\|^2=\|x\|^2+\|y\|^2+2\langle x,y\rangle$$ Thus $$(x,y)\mapsto \frac12(\|x+y\|^2-\|x\|^2-\|y\|^2)$$ is a symmetric positive definite bilinear form on the whole space yielding the norm, so the space with the given norm is an inner product space.