Euclidean geometry and $L_2(\lambda)$ space

Question

Suppose $f,g\in L_2(I,\lambda)$ with $\lambda$ any probability measure and the norm $\| x\|=\sqrt{\langle x, x\rangle}$. Could we have the same geometric properties in this space as in the Euclidean spaces? i.e.

1. if $h$ is the middle point of $f$ and $g$, i.e. $h=\frac{1}{2}(f+g)$, then $\| f-h\|=\|h-g\|$.

2. if $\| x-f\|=\|x-g\|$ and $h$ is the middle point of $f$ and $g$, then $\langle x-h, h\rangle=0$?

Answer 1

The span of two linearly independent vectors in a Hilbert space is isometric to $\mathbb R^2$ or $\mathbb C^2$ (depending on whether you're using real or complex scalars).