# Pointwise Multiplication on Real-Valued Functions is Associative

## Theorem

Let $S$ be a non-empty set.

Let $f, g, h: S \to \R$ be real-valued functions.

Let $f \times g: S \to \R$ denote the pointwise product of $f$ and $g$.

Then:

$\left({f \times g}\right) \times h = f \times \left({g \times h}\right)$

## Proof

 $\, \displaystyle \forall x \in S: \,$ $\displaystyle \map {\paren {\paren {f \times g} \times h} } x$ $=$ $\displaystyle \paren {\map f x \times \map g x} \times \map h x$ Definition of Pointwise Multiplication of Real-Valued Functions $\displaystyle$ $=$ $\displaystyle \map f x \times \paren {\map g x \times \map h x}$ Real Multiplication is Associative $\displaystyle$ $=$ $\displaystyle \map {\paren {f \times \paren {g \times h} } } x$ Definition of Pointwise Multiplication of Real-Valued Functions

$\blacksquare$