Sparse vector ($|\mathrm{supp}x|\le s$)

A vector is s-sparse if it has at most $s$ nonzero coordinates. For $x\in {\mathbb{R}}^N$, its best s-term ${\ell }^1$ approximation error is ${\sigma }_s(x)={\min }_{z\text{ s-sparse}}\Vert x-z{\Vert }_1$, equal to the sum of the absolute values of the coordinates outside the largest $s$.

The support of a finite vector is the set of indices of its nonzero coordinates. It is the support of a function on a finite discrete index set, and its cardinality defines sparsity.

For any field $F$ and vector subspace $W\subseteq F^m$, some $u\in W$ has at least $\dim W$ nonzero coordinates. Choose $u$ with largest support of a vector $S$. Restriction from $W$ to $F^S$ is injective: a nonzero vector in its kernel of a linear map vanishes on $S$, so adding it to $u$ would enlarge $S$. Thus $\dim W\le |S|$. The argument works over finite fields, where assuming a generic vector avoids finitely many hyperplanes would be invalid.