Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $I$ be a set, with the operations, the difference and the constant maps $0$ and $1$ on $\mathbb{R}^{I}$ of Pointwise Operations on Real-Valued Maps §operations and Pointwise Operations on Real-Valued Maps §constants. Let $f,g\in\mathbb{R}^{I}$ and $s,t\in\mathbb{R}$.

¶ $\mathbb{R}^{I}$, with the sum $(f,g)\mapsto f+g$, the product $(f,g)\mapsto fg$, $0$ and $1$, is a commutative ring, and the negative of $f$ in this ring is $-f$.

¶ $(f-g)(p)=f(p)-g(p)$ for every $p\in I$.

¶ $s(tf)=(st)f$, $1f=f$, $(s+t)f=sf+tf$, $t(f+g)=tf+tg$ and $t(fg)=(tf)g$.

¶ $tf=hf$ for the constant map $h$ with value $t$; in particular $0f=0$, $(-1)f=-f$ and $t0=0$.

¶ $tf=0$ if and only if $t=0$ or $f=0$.

¶ For a finite set $A$, a map $F:A\to\mathbb{R}^{I}$ and $p\in I$: $\bigl(\sum_{a\in A}F(a)\bigr)(p)=\sum_{a\in A}F(a)(p)$, the sum on the left being formed in the commutative ring $\mathbb{R}^{I}$ of the clause ring.