Analytische Geometrie, Edition: version 27 Jan 2010 by Otto Mutzbauer

R, compare Ch. 4. A key point in applying direct methods is compactness, compactness of sublevel sets, or compactness of all minimizing sequences. However there are interesting cases where compactness cannot hold in any reasonable topology even through minimizers may exist.

Note V that e°° = M(L)I We denote by E1 the classes of simple n-vectors with first component X00 = 1 and we regard M as a map from the space of linear maps from Rx into RN , or equivalently from the N x n-matrices M(N, n) into A,,,Rn+N By the previous remark we have E'1 = { E AnRn+N I = M(G) for some G in M(N, n)} . We denote also by E'+ the cone from 0 to El. Of course E+ is the cone of all simple n-vectors with positive first component E+ := X00>0}. Rn+N Coo = 1} {CEAn,R'+NCoo =0} > 0} E A, W.

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