Pointwise Operations on Real-Valued Maps Exist and Are Unique

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $I$ be a set, and let $\mathbb{R}^{I}$ be the set of maps from $I$ to $\mathbb{R}$.

¶ For each of the operations $+$ and $\cdot$ of $\mathbb{R}$, written $\ast$, there is exactly one binary operation $\circledast$ on $\mathbb{R}^{I}$ such that $(f\circledast g)(p)=f(p)\ast g(p)$ for all $f,g\in\mathbb{R}^{I}$ and $p\in I$.

¶ There is exactly one map $\mu:\mathbb{R}\times\mathbb{R}^{I}\to\mathbb{R}^{I}$ such that $\mu(t,f)(p)=t\,f(p)$ for all $t\in\mathbb{R}$, $f\in\mathbb{R}^{I}$ and $p\in I$.

¶ There is exactly one map $\nu:\mathbb{R}^{I}\to\mathbb{R}^{I}$ such that $\nu(f)(p)=-f(p)$ for all $f\in\mathbb{R}^{I}$ and $p\in I$.

¶ For every $c\in\mathbb{R}$ there is exactly one $h\in\mathbb{R}^{I}$ such that $h(p)=c$ for all $p\in I$.