Hilbert compacts, compact ellipsoids, and compact extrema

We consider a system of so-called Hilbert compacts $K(H)$ in a Hilbert space $H$; those Hilbert compacts admit a two-sided estimate by compact ellipsoids in $H$. For functionals in $H$, we introduce the notion of a compact extremum achieved at a certain base with respect to the imbedding in $K(H)$. An example of the $K$-extremum of a variational functional in the Sobolev space $W_2^1$ is considered.