# Pointwise Multiplication on Real-Valued Functions is Commutative

## Definition

Let $S$ be a non-empty set.

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

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

Then:

$f \times g = g \times f$

## Proof

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

$\blacksquare$