Broken and conforming polynomial spaces on a matching mesh

Let $\mathcal{T}$ be a matching affine simplicial mesh of $\mathrm{\Omega }$ as defined in Matching affine simplicial meshes and shape regularity. Fix $k\in {\mathbb{N}}_0$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, and an integer $q\ge 1$. On each cell let ${\mathbb{P}}_k(K;{\mathbb{K}}^q)$ denote the restrictions to $K$ of ${\mathbb{K}}^q$-valued polynomials of total degree at most $k$. The broken polynomial space is the finite product $V_h^{\mathrm{b}\mathrm{r}}:={\prod }_{K\in \mathcal{T}}{\mathbb{P}}_k(K;{\mathbb{K}}^q),$ viewed as functions defined almost everywhere cell by cell. For an interior facet $F=K\cap K^{\mathrm{\prime }}$, the jump is the difference of the two polynomial traces, with either fixed ordering of $K,K^{\mathrm{\prime }}$; its vanishing is independent of that ordering. The conforming polynomial space is $V_h^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}:=\{v\in V_h^{\mathrm{b}\mathrm{r}}:v{\mathrm{\mid }}_{F,K}=v{\mathrm{\mid }}_{F,K^{\mathrm{\prime }}}\text{ on every interior facet }F\}\mathrm{.}$ The zero-boundary subspace consists of those $v\in V_h^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ whose polynomial trace is zero on every boundary facet. These are algebraic definitions. The Sobolev inclusions and the zero-boundary closure conclusion are proved separately in the corresponding local theorems on matching piecewise polynomials and zero-boundary values. If $\mathcal{T}=\mathrm{\varnothing }$, then $\mathrm{\Omega }=\mathrm{\varnothing }$ by the mesh definition; the empty product is the zero vector space consisting of the unique empty function, and the trace conditions are vacuous.