Pointwise Multiplication on Complex-Valued Functions is Commutative
Jump to navigation
Jump to search
Definition
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 |
$\blacksquare$