The strict nullspace property

Let $m,n\in \mathbb{N}$ with $m,n\ge 1$, and let $A\in M_{m\times n}(\mathbb{R})={\mathbb{R}}^{m\times n}$. Put $[m]:=\{1,\dots ,m\}$ and $[n]:=\{1,\dots ,n\}$. The library indexes vector coordinates and matrix columns by $n=\{0,\dots ,n-1\}$ and matrix rows by $m=\{0,\dots ,m-1\}$. Throughout this item use the canonical relabeling $j\mapsto j-1$ from $[n]$ to the library's coordinate and column indices, and similarly from $[m]$ for rows. Thus write $u_j:=u(j-1)$ and $A_{ij}:=A(i-1,j-1)$. For $u\in {\mathbb{R}}^n$, define the matrix-vector product by $(Au{)}_i:={\sum }_{j=1}^n{A}_{ij}{u}_j(i\in [m]),$ and define $\ker A:=\{h\in {\mathbb{R}}^n:Ah=0\}\mathrm{.}$ For $h\in {\mathbb{R}}^n$ and $S\subseteq [n]$, let $h_S\in {\mathbb{R}}^n$ be the vector whose $j$th coordinate is $h_j$ for $j\in S$ and $0$ for $j\mathrm{\notin }S$; write $S^c:=[n]\setminus S$. For $u\in {\mathbb{R}}^n$, write $\mathrm{\parallel }u{\mathrm{\parallel }}_1:={\sum }_{j=1}^n\mathrm{\mid }{u}_j\mathrm{\mid },$ the same $p=1$ norm as the library's finite-dimensional norm under this coordinate relabeling. For $s\in \mathbb{N}$ with $1\le s\le n$, the matrix $A$ satisfies the strict nullspace property of order $s$ if $\mathrm{\parallel }{h}_S{\mathrm{\parallel }}_1<\mathrm{\parallel }{h}_{S^c}{\mathrm{\parallel }}_1$ for every nonzero $h\in \ker A$ and every $S\subseteq [n]$ with $\mathrm{\mid }S\mathrm{\mid }\le s$. This condition concerns exact kernel vectors only and says nothing about vectors with $Ah\mathrm{\ne }0$.

The matrix space and its zero-based row and column indices

Matrix-vector multiplication and identity matrices

The finite-dimensional one-norm

Foucart and Rauhut, A Mathematical Introduction to Compressive Sensing, §4.1, Definition 4.1

Bandeira, Ten Lectures and Forty-Two Open Problems in the Mathematics of Data Science, §6.3