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.