# Embedding metric spaces into Hilbert ones

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

• All compact metric spaces embeds into the Hilbert cube, which itself embeds into the Hilbert space. – Aleksei Kulikov Jun 15 at 22:57
• Thanks! Reference? – Aryeh Kontorovich Jun 15 at 23:06
• Honestly, both are Wikipedia:) but the second one is an explicit construction while the first one is: wlog the diameter of $K$ isless than $1$; choose countable dense set $x_1, x_2, \ldots$ and map $x$ to $(d(x, x_1), d(x, x_2), \ldots)$. This is a continuous injection from compact set into Hilbert cube, hence homeomorphism. – Aleksei Kulikov Jun 15 at 23:10
• @AlekseiKulikov This seems to only require separability and boundedness -- is compactness really necessary? – Aryeh Kontorovich Jun 16 at 6:47
• @AryehKontorovich Compactness is not needed, actually every Polish space embeds into the Hilbert cube (even more: a space is Polish iff it is homeomorphic to a $G_\delta$ subset of the Hilbert cube). Boundedness is irrelevant because any metric space is homeomorphic to a bounded metric space – Alessandro Codenotti Jun 16 at 8:23

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 $$j(X)$$).