Talk:Strictly convex space

I see a problem with this definition:

a strictly convex space is one for which, given any two points x and y in the boundary ∂B of the unit ball B of V, the affine line L(x, y) passing through x and y meets ∂B only at x and y.

The problematic case is when the V vector space is over the field of complex numbers. The affine line is a shifted one-dimensional linear subspace, but in the complex case this is topologically a 2-dimensional manifold. So according this definition $\mathbb {C} ^{1}$ with norm $\|x\|:=x{\overline {x}}$ is not strictly convex, because each affine line is $\mathbb {C} ^{1}$ itself, so it meets the unit sphere in more than 2 points. However, this is an inner product space, so it should be stricly convex. 89.135.19.250 (talk) 06:59, 14 March 2013 (UTC)