Attainable null space

Definition

Let $\mathcal{S}$ be the Schwartz space, and let $\mathcal{S}'$ be the space of its tempered distributions, with the topology of convergence against each element of $\mathcal{S}$.

Let $q$ be a continuous seminorm on $\mathcal{S}$.

The attainable null space of $q$ is the set

$$\mathcal{N}_q = \{\, p \in \mathcal{S}' : p = \lim_{k} \chi_k \text{ in } \mathcal{S}' \text{ for some sequence } (\chi_k) \subset \mathcal{S} \text{ with } q(\chi_k) \to 0 \,\}.$$

Let $\mathrm{B} \colon \mathcal{S} \to \mathcal{S}'$ be a positive-semi-definite operator. This means that $\langle \mathrm{B}\varphi, \varphi \rangle$ is real and nonnegative for every $\varphi \in \mathcal{S}$.

The seminorm induced by $\mathrm{B}$ is $|\varphi|_{\mathrm{B}} = \langle \mathrm{B}\varphi, \varphi \rangle^{1/2}$.

For this case, $\mathcal{N}_{\mathrm{B}}$ denotes the attainable null space of $|\cdot|_{\mathrm{B}}$.