Does every compact metric space continuously embed into a Hilbert space (possibly with large distortion)?



Embedding metric spaces into Hilbert ones
You may also send $X$ into a space $L\_2(X,\mu)$ via the Fréchet-Kuratowski isometry $x\mapsto d(\cdot,x)\in C^0(X),\|\cdot\|\_\infty$, followed by the bounded linear inclusion $C^0(X)\to L\_2(X,\mu)$, where $\mu$ is a probability measure on $X$. If $\text{supp}(\mu)=X$ (e.g. $\mu$ is a series of deltas $\sum\_{k=1}^\infty2^{-k}\delta\_{q\_k}$ for a dense set $\{q\_k\}\_{k\ge1}$), then $C^0(X)\to L\_2(X,\mu)$ is injective, and the composition $j:X\to L\_2(X,\mu)$ is a continuous embedding (a homeo of $X$ with $j(X)$).


