Consider the set of compactly supported smooth functions $C^\infty_c(\mathbb{R})$. The dual of this space, $(C^\infty_c(\mathbb{R}))'$ is often referred to as a very large space. Since every $f \in C^\infty_c(\mathbb{R})$ can be mapped to an element in the dual space by integration we know that $C^\infty_c(\mathbb{R}) \subset (C^\infty_c(\mathbb{R}))'$ and that this inclusion is strict since non-regular distributions are known to exist.
But how do we know that the dual space is much larger than the original space? That is, how do we know the non-regular distributions are not rare? Is there a way to quantify how much bigger the dual space is compared to the original space?
Given the inclusion you have included, introduction of the Hahn-Banach Theorem allows for the extension of linear functionals defined on a subspace (like $C_c^\infty(\mathbb{R}))$ to the entire space (like $C_c^\infty(\mathbb{R}))'$) in multiple ways. This theorem underpins the ability to create many distinct distributions from a single function.