Reference finite elements and unisolvent degrees of freedom

Let $d\ge 1$, let $\widehat{K}\subset {\mathbb{R}}^d$ be a nonempty bounded open reference cell, let $q\ge 1$ be an integer, and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. A reference finite element is a triple $(\widehat{K},\widehat{P},\widehat{\mathrm{\Sigma }})$ where $\widehat{P}$ is an $N$-dimensional subspace of $[C(\bar{\widehat{K}};\mathbb{K}){]}^q$ for $N\in {\mathbb{N}}_0$ and $\widehat{\mathrm{\Sigma }}=({\widehat{\sigma }}_1,\dots ,{\widehat{\sigma }}_N)$ is a list of linear functionals on $\widehat{P}$ for which $J_{\widehat{\mathrm{\Sigma }}}:\widehat{P}\to {\mathbb{K}}^N,J_{\widehat{\mathrm{\Sigma }}}(p):=({\widehat{\sigma }}_1(p),\dots ,{\widehat{\sigma }}_N(p)),$ is bijective. The functionals are the degrees of freedom and the triple is unisolvent. Its shape functions are the unique dual basis ${\widehat{\varphi }}_1,\dots ,{\widehat{\varphi }}_N\in \widehat{P}$ satisfying ${\widehat{\sigma }}_i({\widehat{\varphi }}_j)={\delta }_{ij}$.

If $X$ is a normed vector space of functions on $\widehat{K}$ containing $\widehat{P}$, fix for each $i$ a bounded linear extension of ${\widehat{\sigma }}_i$ to $X$, and denote this chosen extension by the same symbol. The reference interpolation operator associated with these chosen extensions is $\widehat{I}:X\to \widehat{P},\widehat{I}v:={\sum }_{i=1}^N{\widehat{\sigma }}_i(v){\widehat{\varphi }}_i\mathrm{.}$ The definition of a reference finite element does not assert that any particular list of functionals is unisolvent or bounded on a given Sobolev space; those are separate proof obligations. For $N=0$, $\widehat{P}=\{0\}$, ${\mathbb{K}}^0=\{0\}$, the empty list gives the unique bijection between these zero spaces, and the shape-function list is empty. The corresponding interpolation formula is the empty sum, hence the zero map into $\{0\}$.