Linear Dugundji extension remains topological

For a metric space $X$, a nonempty closed subset $A\subseteq X$ and a locally convex topological vector space $L$, every continuous $f:A\to L$ has a continuous extension $F:X\to L$ with image contained in the convex hull of $f[A]$ (the convex-valued Dugundji extension theorem). The catalogue's intended stronger claim, that the extension can be chosen by a single linear operator $C(A,L)\to C(X,L)$ continuous for uniform convergence on compact sets, is preserved but not established here.

No metamathematical independence result is a proof supplier for either form; the original paper's arguments cover the convex-valued extension and the bounded scalar supremum-norm version, not the compact-open operator statement. The draft target is Dugundji's extension theorem in its linear form ‡.

• Orientation only. Nothing on this page depends on this remark, and the topological extension theory is not part of the Gelfand proof spine.
• Open obligation for the catalogue. Either obtain a complete source or proof of the compact-open operator form, or narrow the catalogue target to the proved formulation; the choice cost of the metric paracompactness input must be recorded in that repair.