Real singular cochains identify with functions on the supplied simplex basis

For $a:S\to \mathbb{R}$ define $L_a(\sum r_s[s])=\sum r_{s}a(s)$ and $A_a(\sum n_s[s])=\sum n_{s}a(s)$. Unique finite coefficients make both sums well-defined; finite distributivity proves real linearity of $L_a$ and additivity of $A_a$. Scalar multiplication of additive maps is pointwise.

Evaluation sends $L_a$ and $A_a$ back to $a$ because their value on $[s]$ is $a(s)$. Conversely every real-linear $L$ satisfies $L(\sum r_s[s])=\sum r_{s}L([s])$, and every additive $A$ satisfies $A(\sum n_s[s])=\sum n_{s}A([s])$, including negative integers by additive inverses. Thus evaluation and extension are inverse in both cases and are real-linear. There is no finite-support condition on $a$.

Write $D[t]={\sum }_s{d}_{st}[s]$ with each column finite, and let $D_{\mathbb{R}}$ use the same formula over $\mathbb{R}$. Both $L_a{D}_{\mathbb{R}}$ and $A_{a}D$ have value ${\sum }_s{d}_{st}a(s)$ on $[t]$, so the identifications commute with these maps. A signed face sum is such a finite column, even when repeated faces combine. Zero columns and the degree-zero boundary give zero values.

If $S=\mathrm{\varnothing }$, all three spaces are zero, with the unique empty function. If $S$ is a singleton, evaluation is the usual identification with $\mathbb{R}$; in negative singular degrees the groups are zero by convention. All inverse maps have specified formulas on the supplied formal generators, so no representative or basis selection and no AC is used.