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Example text

T)1 < . (t)l· til It appears that every lk_ or lk-space may be considered as an 00 1 1- I or loo-space resp. For if x:={x(t)}7 E 1~, then the sequence {~r}7 defined -by ~ i=I,2, ... 10) is a vector of II with the same norm. The other way round, with the help of an opposite process every vector of II may be identified with a vector of 1~ possessing the same norm. 26 A similar relation may be constructed between 100 and 1:. This means that all properties of the 1 1 - and loo-space simply can be transferred to the l~- and l:-spaces resp.

1) possesses a solution x E 1 n+; this solution x or a solution (x,y) E In x 1m. 5) will be called dual feasible, feasible (P- and D-feasib1e) when the LP-system is P- and D- feasible, virtually feasible (po_ and D-feasib1e) when the LP-system is po_ and D-feasible. 5~5) poccesses a solution + 31 (u,v) E 1: x 1: such that (ul/n'v l / n ) E 1:+ x int(l:+); this solution is called dual-regular. The notions virtually primal regular (po-regular), regular and virtuallyregular may be introduced in a similar manner as the corresponding notions with respect to the feasibility.

The latter is essential to the argumentation of the next theorem. 8 Theorem. 'V For anLP-system (P- or D-directed; f(t) .... f, t .... 8. I) ~ has a solution. 2) 44 has a solution. 3) has a solution. Proof. (a) Let (x,y) E In:m be a P-feasible solution for some x(O). 4) 1 t Y > - E (-) y(t) = T t= 1 P has a solution for every T > I. 7) g(T) E C, T>I. '" The supposition f(t) .... f, '" g (T) .... f, t .... 8) 00 From the definition of C it follows ( 8:174) that this set is '" closed. 8) imply f E C.

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