Pointwise Multiplication on Complex-Valued Functions is Commutative

From ProofWiki
Jump to navigation Jump to search

Definition

Let $S$ be a non-empty set.

Let $f, g: S \to \C$ be complex-valued functions.

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


Then:

$f \times g = g \times f$


That is, pointwise multiplication of complex-valued functions is commutative.


Proof

\(\ds \forall x \in S: \, \) \(\ds \map {\paren {f \times g} } x\) \(=\) \(\ds \map f x \times \map g x\) Definition of Pointwise Multiplication of Complex-Valued Functions
\(\ds \) \(=\) \(\ds \map g x \times \map f x\) Complex Multiplication is Commutative
\(\ds \) \(=\) \(\ds \map {\paren {g \times f} } x\) Definition of Pointwise Multiplication of Complex-Valued Functions

$\blacksquare$