Cellwise L2 projection relation onto broken polynomials

Let $K$ be a cell in the one-dimensional interval or $d\ge 2$ matching affine-simplex branch of Upwind discontinuous Galerkin discretization of constant transport, and fix an integer degree $p\ge 0$. Let $(K,\mathcal{A},\mu )$ be a finite measure space as in Measure spaces, let ${\mathcal{L}}^2(\mu )$ have the raw measurable-function meaning of The function space ${\mathcal{L}}^p(\mu )$ for $0<p<\mathrm{\infty }$, and let $W={\mathbb{P}}_p(K)\subseteq {\mathcal{L}}^2(\mu )$. For $u\in {\mathcal{L}}^2(\mu )$ define its cellwise L2 projection relation to be the set ${\mathcal{P}}_K(u)=\{p\in {\mathbb{P}}_p(K):{\int }_K(u-p)wd\mu =0\text{ for every }w\in {\mathbb{P}}_p(K)\}\mathrm{.}$ For a mesh, the broken relation is the product of these cell relations. The measure $\mu$ is required to give positive measure to every nonempty relatively open subset of $K$ when existence and uniqueness are claimed. This definition does not itself assert that the set is a singleton. Whenever a later result proves that the relation has a unique member, write ${\mathrm{\Pi }}_{K}u$ for that member.