Attainable null space

Summary

For a continuous seminorm $q$ on the Schwartz space $\mathcal{S}$, the attainable null space $\mathcal{N}_q$ is a set of tempered distributions.

A distribution $p$ in $\mathcal{S}'$ lies in $\mathcal{N}_q$ when some sequence $(\chi_k)$ in $\mathcal{S}$ converges to $p$ in $\mathcal{S}'$ and has $q(\chi_k)\to 0$.

Because $q$ is a seminorm, $\mathcal{N}_q$ is a linear subspace of $\mathcal{S}'$: sums and scalar multiples of such sequences are again such sequences.

So $\mathcal{N}_q$ collects the distributions that test functions of vanishing $q$-size can approximate.

For a positive-semi-definite operator $\mathrm{B}$ from $\mathcal{S}$ to $\mathcal{S}'$, the set $\mathcal{N}_{\mathrm{B}}$ is the case where $q$ is the seminorm induced by $\mathrm{B}$.