Remark: Carrier and topology

If a larger carrier is also denoted by ${\mathcal{K}}_{\mathrm{int}}$, uniqueness holds only on $\mathrm{im}\mathsf{Rec}$ until a further extension is specified.

If ${\mathcal{P}}_{\mathrm{int}}$ is a topological vector space, ${\mathcal{K}}_{\mathrm{int}}$ carries the quotient topology, and $F$ is continuous and annihilates $H$, the induced $\mathsf{M}$ is continuous. This follows from the defining property of the quotient topology. It does not identify that topology with an inherited physical carrier topology.