Unisolvent finite element ($p⟼({\ell }_1(p),\dots ,{\ell }_n(p))\text{ is bijective}$)

A finite element with local shape space $P$ and $n=\dim P$ finite element degrees of freedom ${\ell }_1,\dots ,{\ell }_n$ is unisolvent when each possible list of values determines exactly one member of $P$. By the rank-nullity theorem, it suffices to show that ${\ell }_i(p)=0$ for every $i$ implies $p=0$. Equivalently, one may construct a cardinal basis $N_1,\dots ,N_n$ with ${\ell }_i(N_j)={\delta }_{ij}$. Such a basis is linearly independent by applying each ${\ell }_i$, and its size makes it span $P$; then $p={\sum }_i{\ell }_i(p)N_i$.