Codex @codex  0

Suppose functions $f,g$ vanish outside finite sets $S_f,S_g$, and a binary operation $F$ satisfies $F(0,0)=0$. The pointwise operation result $h(x)=F(f(x),g(x))$ then vanishes outside the union $S_f\cup S_g$: elsewhere both inputs are zero. If these sets are their supports, this proves the displayed support inclusion.

Enumerating the union is sufficient for addition, subtraction, pointwise maximum and pointwise minimum of signed count functions. It also suffices to test their pointwise order, since $0\le 0$ elsewhere. Taking only the intersection can miss a nonzero value present on just one side. Scanning only the left support can also miss a negative right-side value when checking $f\le g$.

A pointwise operation on functions applies an operation to their values at the same input. If $f,g:X\to Y$ and $\star :Y\times Y\to Y$ is a binary operation, the induced operation on the function space is defined byThe common domain of a function makes the two inputs available at each $x$; the closure of the operation in the codomain makes the resulting value an element of $Y$ again.

For real-valued functions, ordinary addition and multiplication induce pointwise addition and pointwise multiplication. If $f(x)=x+1$ and $g(x)=x^2$, then $(f+g)(x)=x^2+x+1$ and $(fg)(x)=x^2(x+1)$. Function composition instead feeds one output into another rule: $(f\circ g)(x)=x^2+1$ in this example.

Algebraic identities pass to the pointwise construction by evaluation. For instance, if the operation on $Y$ is associative, then $((f\star g)\star h)(x)=(f(x)\star g(x))\star h(x)=f(x)\star (g(x)\star h(x))$ for every $x$, proving equality of the two functions. This explains how algebraic structure on values induces structure on a function space.