Commutative and Associative Product on Space of Distributions does not Exist

Theorem
Let $\map {\DD'} \R$ be the distribution space.

Let $\alpha \in \map {C^\infty} \R$ be a smooth function.

Let $\circ$ be a product operation on $\map {\DD'} \R$.

Suppose:


 * $\forall T \in \map {\DD'} \R : \forall \alpha \in \map {C^\infty} \R : \alpha \circ T := \alpha \cdot T$

where $\cdot$ stands for multiplication of distribution by a smooth function.

Suppose $\circ$ is commutative and associative.

Then $\circ$ does not exist.

Proof
there is such a product.

Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.

We also have that $\ds \PV \frac 1 x$ is a distribution.

Let $\phi, \psi \in \map \DD \R$ be test functions.

Then:

Here $\mathbf 1$ is the distribution associated to the real function defined by:


 * $\mathbf 1 : \R \to 1$

Consider the product $\ds \delta \cdot x \cdot \PV \frac 1 x$ in the distributional sense, where $\delta$ and $\PV \frac 1 x$ are distributions, and $x$ is a smooth function.

Altogether, we have that:


 * $\ds \paren {\delta \circ x} \circ \PV \frac 1 x \ne \delta \circ \paren {x \circ \PV \frac 1 x}$

Hence, the associativity is violated, and we have reached a contradiction.