During my research I have stumbled upon the following issue concerning infinite systems of linear equations. I do not have much practice in such settings, so I am asking you whether the following connotes you something --- perhaps there is already some developed theory of such or similar systems.
We work in $\mathbb{R}^\mathbb{N}$ endowed with the pointwise addition and multiplication by scalars. Let $(\varphi_n\colon\ n\in\mathbb{N})$ be a sequence of functionals such that for each $n\in\mathbb{N}$ there exist $k_1,\ldots,k_m\in\mathbb{N}$ and $\alpha_1,\ldots,\alpha_m\in\mathbb{R}\setminus\{0\}$ such that for every $x\in\mathbb{R}^\mathbb{N}$ we have $\varphi_n(x)=\alpha_1x(k_1)+\ldots+\alpha_mx(k_m)$ (so $\varphi_n$ is a linear combination of some Dirac's delta on $\mathbb{N}$) and $|\alpha_1|+\ldots+|\alpha_m|=1$. Let $G$ be a finite subset of $\mathbb{N}$, $g\colon G\to\mathbb{R}$ and $\varepsilon>0$.
Does there exist $x\in\mathbb{R}^\mathbb{N}$ satisfying the following conditions:
1) there exists $n\in\mathbb{N}$ such that for every $l\ge n$ we have $\varphi_l(x)=0$;
2) for every $k\in G$ we have $|g(k)-x(k)|<\varepsilon$?
Do you know whether such or similar problems have been already considered somewhere? Maybe they already have a name, so I can easier look for some results? I'll be grateful for any hint!