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

\(\, \displaystyle \forall x \in S: \, \) \(\displaystyle \left({\left({f \times g}\right) \times h}\right) \left({x}\right)\) \(=\) \(\displaystyle \left({f \left({x}\right) \times g \left({x}\right)}\right) \times h \left({x}\right)\) Definition of Pointwise Multiplication of Real-Valued Functions
\(\displaystyle \) \(=\) \(\displaystyle f \left({x}\right) \times \left({g \left({x}\right) \times h \left({x}\right)}\right)\) Real Multiplication is Associative
\(\displaystyle \) \(=\) \(\displaystyle \left({f \times \left({g \times h}\right)}\right) \left({x}\right)\) Definition of Pointwise Multiplication of Real-Valued Functions

$\blacksquare$