Example of an infinite-dimensional geodesic NPC Space I just started reading Ballmann's book on non-positive curvature spaces.  In it most, non-linear, examples of NPC spaces are negatively curved manifolds or specific graphs/discrete metric spaces, or buildings.
So, to gain intuition, what is a "down-to-earth"/interesting example of a metric space $(X,d)$ which:

*

*Has non-positive curvature (in the sense of Ballmann)

*X is not a topological vector space but it is bi-Lipschitz equivalent to an infinite-dimensional separable Hilbert space $H$.

*"X is not only a toy example but is interested in other areas of math...has reasonable "roots""


I'll briefly sketch the construction of an infinite dimensional hyperbolic space mentioned by Moishe. Let $H$ be an infinite dimensional separable Hilbert space, and consider the space $V:=\mathbb R\oplus H$. We can equip $V$ with an inner product $\langle(\lambda_1,h_1),(\lambda_2,h_2)\rangle_V:=\lambda_1\lambda_2-\langle h_1,h_2\rangle_H$, turning it into a separable Hilbert space, with quadratic form, $Q(\cdot)=\|\cdot\|^2_V$. $V$ has a natural cone, $C=\{(\lambda,x)\in V: \|x\|_H<\lambda\}$. We can consider a slice of this cone $\mathcal H:=\{v\in C: Q(v)=1\}$. Finally, we equip $\mathcal H$ with a metric, $d(u,v):=\operatorname{arcosh}(\langle,u,v \rangle).$ The metric space $(\mathcal H,d)$ is known as the hyperboloid model of infinite dimensional hyperbolic space, and is the natural extension of Minkowski's model of hyperbolic space in finite dimensions.
