Curvature of space of paths into a NPC space

Question

Let $(X,d)$ be a complete separable geodesic (thus connected and path connected) metric space of non-positive curvature (in the sense of Ballmann) 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...

Answer 1

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.