I have some problem, which I thought, is already solved on this website, but I couldn't find it.
Let $H$ be a real, nonzero Hilbert space, and let $E$ be an orthonormal basis of $H$. Denote $V:=\left\{ h\in H \ : \ \Vert h-e\Vert =\Vert h-e'\Vert, \ \ \ e,e'\in E\right\}$. Show that $V=\left\{0\right\}$ iff $\dim H=\infty$.
Assume $\dim H =n< +\infty$.
Define $h= \sum_{k=1}^n e_k \ne 0$.
Then: $$\left\|\sum_{k\ne i} e_k\right\|^2 = n - 1 = \left\|\sum_{k\ne j} e_k\right\|^2, \quad \forall j\in\{1, \ldots, n\}$$
Hence $h \in V$.
Conversely, assume $0 \ne h \in V$.$\newcommand\inner[2]{\left\langle #1, #2 \right\rangle}$
For any $e, e' \in E$ we have: $$\|h - e\|^2 = \|h - e'\|^2 \implies \inner{h-e}{h-e} = \inner{h-e'}{h-e'}$$
From here we obtain $$\Re\inner{h}{e} = \Re\inner{h}{e'}$$
Let $E = \{e_j : j \in J\}$. $$h = \sum_{j \in J}\inner{h}{e_j} e_j \implies \|h\|^2 = \sum_{j \in J}\left|\inner{h}{e_j}\right|^2 \geq \sum_{j \in J}\left(\Re\inner{h}{e_j}\right)^2$$
where $\Re\inner{h}{e_j}$ are equal for every $j \in J$.
$h \ne 0$ implies $\|h\| \ne 0$ so $\inner{h}{e_i} \ne 0$, for some $i \in J$. $H$ is real so in fact we have $\Re\inner{h}{e_j} \ne 0$.
Since $\|h\| < +\infty$, we get that $J$ must be finite.
Hence, $\dim H < +\infty$.