Space of linear combinations of bounded functions and $l_1$ Banach space

Question

Let $I$ be an at most countable and non-empty set and $(h_i)_{i\in I}$ be a family of bounded functions $h_i:X\rightarrow \mathbb{R}$ with $\lVert h_i \rVert_{\infty}\leq 1$. In addition, $(w_i)_{i\in I}$ is an $\mathbb{R}$-valued family and
$l_1(I):=\{(w_i)_{i\in I}: \lVert (w_i)_{i\in I} \rVert_{l_1(I)} < \infty\}\,,$
where
$\lVert (w_i)_{i\in I} \rVert_{l_1(I)}:=\sum_{i\in I}|w_i|$.
What would be the space of $f=\sum_{i\in I} w_i h_i$ ? and properties?