Proposition: Explicit failure witnesses

If (eq:extensionbound) fails for a compact $K$, then there exist tests

such that

Any such sequence excludes a distributional extension of $b^{\circ }$ to $X$.

Proof. Failure of the estimate means that for each $n$ the bound with order $n$ and constant $n$ fails. Choose a nonzero test ${\psi }_n$ with

The seminorm $p_n({\psi }_n)$ is positive because it includes the zeroth derivative. Dividing by it gives (eq:failuresequence).

If an extension existed, on $K$ it would satisfy $|⟨b,\varphi ⟩|\le C{p}_k(\varphi )$ for some $C,k$. For $n\ge k$, the sequence obeys $p_k({\varphi }_n)\le p_n({\varphi }_n)\le 1$, whereas its pairings exceed $n$. This contradicts the estimate for all sufficiently large $n$.