Polynomial derivative and face norms are controlled on a reference simplex

The cell expression $N_0(v):=\mathrm{\parallel }v{\mathrm{\parallel }}_{L^q({\widehat{S}}_d)}$ is a norm on $\widehat{P}$. If it is zero, then for finite $q$ the continuous polynomial vanishes almost everywhere, and for $q=\mathrm{\infty }$ it vanishes essentially everywhere. In either case, if it were nonzero at an interior point, continuity would make it bounded away from zero on an interior ball of positive measure, a contradiction. It therefore vanishes on the open simplex and hence is the zero polynomial, since its finite Taylor expansion at any interior point is exact.

If $\widehat{P}=\{0\}$, all displayed estimates and functional bounds hold immediately, so assume $\dim \widehat{P}>0$. The expression $N_1(v):=\mathrm{\parallel }v{\mathrm{\parallel }}_{L^p({\widehat{S}}_d)}+\mathrm{\mid }v{\mathrm{\mid }}_{W^{m,p}({\widehat{S}}_d)}+\mathrm{\parallel }v{\mathrm{\parallel }}_{L^p(\widehat{F})}$ is a norm: it is nonnegative, homogeneous and subadditive, and $N_1(v)=0$ forces the cell $L^p$ norm to vanish, so the preceding argument gives $v=0$. For real coefficients apply [F1], and for complex coefficients apply [F2], to $N_0$ and $N_1$; norm equivalence gives $N_1(v)\le C{N}_0(v)$. Taking the maximum over the finitely many facets proves their uniform estimate. By [F3], any degree-of-freedom functional is linear on $\widehat{P}$; it is bounded for $N_1$ by finite-dimensional norm equivalence and hence by $N_0$.

If $\widehat{P}=\{0\}$, the second estimate is immediate. If $m=0$, the second estimate is the derivative part of step 2.1 with its order there set to $\mathrm{\ell }$. If $m\ge 1$, let ${\mathbb{P}}_{m-1}$ be the polynomials of degree at most $m-1$. The seminorm $\mathrm{\mid }v{\mathrm{\mid }}_{W^{m,q}}$ vanishes exactly on $\widehat{P}\cap {\mathbb{P}}_{m-1}$: every monomial of degree at least $m$ has a nonzero derivative of order $m$, and every derivative of order $m$ kills lower-degree monomials. It is therefore a norm on the finite- dimensional quotient $\widehat{P}\mathrm{/}(\widehat{P}\cap {\mathbb{P}}_{m-1})$. If this quotient is zero, then the target seminorm vanishes on all of $\widehat{P}$ because $\mathrm{\ell }\ge m$, so the estimate is immediate. Otherwise add the target seminorm $\mathrm{\mid }v{\mathrm{\mid }}_{W^{\mathrm{\ell },p}}$ to the quotient norm; it is well-defined because it also vanishes on $\widehat{P}\cap {\mathbb{P}}_{m-1}$. The sum is another norm. Applying [F1] or [F2] to these two norms bounds the target seminorm by the order-$m$ seminorm. This proves the second estimate.