Distribution Space over Smooth Functions is Unitary Module

Theorem
The distribution space over smooth functions is a unitary module.

Proof
Let $\phi \in \map \DD {\R^d}$ be a test function.

Hence:


 * $\alpha \paren {T_1 + T_2} = \alpha T_1 + \alpha T_2$

Hence:


 * $\paren {\alpha_1 + \alpha_2} T = \alpha_1 T + \alpha_2 T$

Hence:


 * $\paren {\alpha \beta} T = \alpha \paren {\beta T}$

Let $\mathbf 1 \in \map {C^\infty} {\R^d}$ be a constant mapping such that:


 * $\R^d \stackrel {\mathbf 1} {\mapsto} 1$

Then:

Hence:


 * $1 \cdot T = T$