Geodesic
Hilbert compacts, compact ellipsoids, and compact extrema
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 29 (2008), pp. 165-175

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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. 
@article{CMFD_2008_29_a8,
     author = {I. V. Orlov},
     title = {Hilbert compacts, compact ellipsoids, and compact extrema},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {165--175},
     publisher = {mathdoc},
     volume = {29},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2008_29_a8/}
}
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I. V. Orlov. Hilbert compacts, compact ellipsoids, and compact extrema. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 29 (2008), pp. 165-175. http://geodesic.mathdoc.fr/item/CMFD_2008_29_a8/