---
tag: RK2N3QP4FQ
kind: concept
name: Attainable null space
msc: [46-XX]
specializes: [00YC1D652K]
sources:
  - arxiv: 2610.14906v1
    locator: Definition 3.1
    mention: ax://paper/2610.14906#2P92
    licence: cc-by
    note: the paper that states the item (corpus arxiv AX9.2)
history:
  notes: The source is arXiv:2610.14906v1 (2026-10), by Michael Unser. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it.
inventory: [6YTCGV56SP]
uses: [FP5GVCN8GV, CG9JEATW3Q, AES2T5JPFP]
---

## Summary

For a continuous [[seminorm|FP5GVCN8GV]] $q$ on the [[Schwartz space|CG9JEATW3Q]] $\mathcal{S}$, the **attainable null space** $\mathcal{N}_q$ is a set of [[tempered distributions|AES2T5JPFP]].

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|00YC1D652K]] 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 $\mathrm{B}$.