Pointwise Operations on Real-Valued Maps

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 $f,g\in\mathbb{R}^{I}$ and $t\in\mathbb{R}$, the sum $f+g$, the product $fg$, the scalar multiple $tf$ and the negative $-f$ are the elements of $\mathbb{R}^{I}$ with

$$(f+g)(p)=f(p)+g(p),\qquad (fg)(p)=f(p)\,g(p),\qquad (tf)(p)=t\,f(p),\qquad (-f)(p)=-f(p)\qquad(p\in I),$$

namely the values at $(f,g)$, $(f,g)$, $(t,f)$ and $f$ of the operations and maps of Pointwise Operations on Real-Valued Maps Exist and Are Unique §binary, Pointwise Operations on Real-Valued Maps Exist and Are Unique §scalar-multiple and Pointwise Operations on Real-Valued Maps Exist and Are Unique §negative. The product is written by juxtaposition only, never with a dot. The difference is $f-g=f+(-g)$.

¶ For $c\in\mathbb{R}$, the constant map with value $c$ is the $h\in\mathbb{R}^{I}$ with $h(p)=c$ for all $p\in I$, given by Pointwise Operations on Real-Valued Maps Exist and Are Unique §constant. Where an element of $\mathbb{R}^{I}$ is required, $0$ and $1$ denote the constant maps with values $0$ and $1$.