Pointwise Addition on Complex-Valued Functions is Associative
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Theorem
Let $S$ be a set.
Let $f, g, h: S \to \C$ be complex-valued functions.
Let $f + g: S \to \C$ denote the pointwise sum of $f$ and $g$.
Then:
- $\paren {f + g} + h = f + \paren {g + h}$
That is, pointwise addition on complex-valued functions is associative.
Proof
\(\ds \forall x \in S: \, \) | \(\ds \map {\paren {\paren {f + g} + h} } x\) | \(=\) | \(\ds \paren {\map f x + \map g x} + \map h x\) | Definition of Pointwise Addition of Complex-Valued Functions | ||||||||||
\(\ds \) | \(=\) | \(\ds \map f x + \paren {\map g x + \map h x}\) | Complex Addition is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {f + \paren {g + h} } } x\) | Definition of Pointwise Addition of Complex-Valued Functions |
$\blacksquare$