My question is if a specific metric-space is complete, respectively under which conditions it is complete. I am rather a newby, but hope that the question is understandable.
The metric-space is defined by the set of initial conditions of a system of ordinary differential equations, and a distance measure with respect to their solutions.
Let $\frac{d}{dt}z=f(z,t)$, $z\in\mathbb{Z}\subseteq\mathbb{R}^n$, be a system of ordinary differential equations, with $f$ continuous in $z$ and $t$. Assume that for every initial condition $z(0)=\bar{z}\in\bar{\mathbb{Z}}\subseteq\mathbb{Z}$ there exists a unique and continuous in time solution denoted by $\zeta(t,\bar{z})$ for $t\in\left[0,\infty\right)$.
Given this system of ODEs, I defined the metric-space $(\bar{\mathbb{Z}}, d)$, with $d$ given by $$d(\bar{z}_1,\bar{z}_2)=\int_0^\infty w(t)\left|h(\zeta(t,\bar{z}_1))-h(\zeta(t,\bar{z}_2))\right|^p dt.$$ Here, $h:\mathbb{Z}\rightarrow\mathbb{R}$ is some continuous function, $p\in\left[1,\infty\right)$ and $w:\left[0,\infty\right)\rightarrow(0,\infty)$ some weight. Assume that for every $\bar{z}_1,\bar{z}_2\in\bar{\mathbb{Z}}$ it is given that $0\leq d(\bar{z}_1,\bar{z}_2)<\infty$, and $d(\bar{z}_1,\bar{z}_2)=0\Rightarrow \bar{z}_1=\bar{z}_2$.
I would like to prove that if $\bar{\mathbb{Z}}$ is a closed sub-space of the Euclidean space, $(\bar{\mathbb{Z}}, d)$ is complete.
I tried to prove this by picking some (fixed) $\hat{z}\in\bar{\mathbb{Z}}$ and by defining another space $(K,d_k)$, with $K=\{k:\left[0,\infty\right]\rightarrow\mathbb{R}:k(t)=h(\zeta(t,\bar{z}))-h(\zeta(t,\hat{z})),\bar{z}\in\bar{\mathbb{Z}}\}$, and $d_k(k_1,k_2)=\|k_1-k_2\|_{L_p}^p$ defined by $$\|k_1-k_2\|_{L_p}^p=\int_0^\infty w(t)|k_1(t)-k_2(t)|^pdt.$$ Intuitively, I would guess that $(\bar{\mathbb{Z}}, d)$ is complete if $(K, d_k)$ is complete. Since $(K, d_k)$ is a subspace of the w-weighted $L_p$ space which is complete, I "only" would have to show that $(K,d_k)$ is closed. However, here I got lost...
Is $(\bar{\mathbb{Z}}, d)$ complete for $\bar{\mathbb{Z}}$ being a closed sub-space of the Euclidean space, and, if not, are there some ``mild'' assumptions I could make such that it is complete (I know, mild is not well defined...)?