Let $(X,d)$ be a complete separable geodesic *(thus connected and path connected)* metric space of non-positive curvature (in the sense of [Ballmann](https://people.mpim-bonn.mpg.de/hwbllmnn/archiv/NPC0606.pdf#page=3)) and fix some $x\_0\in X$. Let $C\_0(I,X)$ denote the set of continuous functions $f:[0,1]\rightarrow X$ satisfying
$$
f(0) = x\_0 \mbox{ and } \sup\_{x\in X}\, d(f(x),x\_0) < \infty.
$$
Define the metric $D$ on $C\_0(X)$ by
$$
D(f,g)\,:=\, \sup\_{x\in X}\, d(f(x),g(x))
,
$$
and note that $D$ is always finite.


Is $C\_0(I,X)$ a complete and separable space of non-positive curvature? I mean this is true if $X$ is Banachian but I don't know about the general case...



Curvature of space of paths into a NPC space
Maybe I'm missing something. But it looks like if you take $X = \mathbb{R}$ than $C\_0(I, X)$ is a set of continuous functions on $[0,1]$ with $0$ at $0$. And than it's universal aka (almost) contains an isometric copy of every finite metric space. And NPC is very restrictive in terms of what $4$-point subsets are allowed.


PS: maybe you can consider an $L\_2$-metric
$$D\_{L\_2}(f,g)\,:= \sqrt{\int\_{[0,1]} \big(d(f(x),g(x))\big)^2}$$ and than is will be NPC. But it will not be complete.


