Let $Y$ be a compact, separable metric space and $X=C(Y)$ Banach space. There are many criteria for a linear subspace $Z\subseteq X$ to be dense; notably the [Stone-Weierstraß](https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem) theorem. 


Are there theorems giving conditions on non-linear subsets $Z$ of $X$ to be [ε-dense?](https://math.stackexchange.com/questions/651090/whats-the-difference-between-%CE%B5-dense-and-dense) Particularly, if I know that 


* $Z$ comprises *"partial sums"* of at-most $N$ elements; *i.e.*,
$$
Z=\left\{
k+\sum\_{i=0}^N k\_iy\_i:\, y\_i \in \tilde{Y}, k,k\_i \in \mathbb{R}
\right\},
$$
for some positive integer $N>0$ and some subset $\tilde{Y}\subseteq Z\subseteq C(Y)$,
* $\operatorname{span}(Z)$ is dense in $X$?



Criteria for $\epsilon$-Density
If your set $Z$ is $\epsilon$-dense, then it is also dense! This is in fact so for every normed space $X$, and for every subset $Z\subseteq X$ which is positively homogeneous in the sense that $\alpha Z = Z$, for every scalar $\alpha>0$. The reason is as follows, for any point $x$ in $X$, and any $\alpha>0$, one has that
 $$
 d(\alpha x,Z) = d(\alpha x,\alpha Z) = \alpha \,d(x,Z),
 $$
 where $d$ is the distance function.


Thus, if $Z$ is $\epsilon$-dense, one has for every $x$ in $X$, and every natural number $n>0$, that
 $$
 d(x,Z) = (1/n)\,d(nx,Z) \leq (1/n)\varepsilon,
 $$
 from where you deduce that $d(x,Z)=0$


