Test Function Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space

Theorem
Let $\map \DD {\R^d}$ be the test function space.

Let $\struct {\C, +_\C, \times_\C}$ be the field of complex numbers.

Let $\paren +$ be the pointwise addition of test functions.

Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of test functions over $\C$.

Then $\struct {\map \DD {\R^d}, +, \, \cdot \,}_\C$ is a vector space.

Proof
Let $f, g, h \in \map \DD {\R^d}$ be test functions with the compact support $K$.

Let $\lambda, \mu \in \C$.

Let $\map 0 x$ be a real-valued function such that:


 * $\map 0 x : \R^d \to 0$.

Let us use real number addition and multiplication.

$\forall x \in \R^d$ define pointwise addition as:


 * $\map {\paren {f + g}} x := \map f x +_\C \map g x$.

Define pointwise scalar multiplication as:


 * $\map {\paren {\lambda \cdot f}} x := \lambda \times_\C \map f x$

Let $\map {\paren {-f} } x := -\map f x$.

Closure Axiom
By Sum Rule for Continuous Complex Functions, $f + g \in \map \DD {\R^d}$

Commutativity Axiom
By Pointwise Addition on Complex-Valued Functions is Commutative, $f + g = g + f$

Associativity Axiom
By Pointwise Addition is Associative, $\paren {f + g} + h = f + \paren {g + h}$.

Also see

 * Complex Vector Space is Vector Space
 * Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space