Pointwise Multiplication on Real-Valued Functions is Associative

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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:

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


That is, pointwise multiplication on real-valued functions is associative.


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

$\blacksquare$