Tempered Distribution Space is Proper Subset of Distribution Space

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

Let $\map {\SS'} \R$ be the tempered distribution space.

Then $\map {\SS'} \R$ is a proper subset of $\map {\DD'} \R$:


 * $\map {\SS'} \R \subsetneqq \map {\DD'} \R$

Proof
By Convergence of Sequence of Test Functions in Test Function Space implies Convergence in Schwartz Space we have that $\map {\SS'} \R \subseteq \map {\DD'} \R$.

Consider the real function $\map f x = e^{x^2}$.

We have that:


 * Real Power Function for Positive Integer Power is Continuous
 * Exponential Function is Continuous/Real Numbers
 * Composite of Continuous Mappings is Continuous

Thus, $f$ is a continuous real function.

Also:


 * $\forall x \in \R : e^{x^2} < \infty$

Hence, $f$ is locally integrable.

By Locally Integrable Function defines Distribution, $T_f \in \map {\DD'} \R$.

$T_f$ is a tempered distribution.

We have that $e^{-x^2}$ is a Schwartz test function.

Then:

Hence, $\map {T_f} {e^{-x^2} } \notin \R$.

This is a contradiction.

Therefore, $T_f \notin \map {\SS'} \R$ while at the same time $T_f \in \map {\DD'} \R$.