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Partial Order, Linear Functional, Vector Space and Subspace

Let be a vector space and a subset of such that implies and for Define a partial order on by defining to mean .
A linear functional on is said to be positive (with respect to ) if for . Let be any subspace of with the property that for each there is an with . Assume that , where Then each positive linear functional on can be extended to a positive linear functional on .
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Partial Order, Linear Functional, Vector Space and Subspace are investigated. The solution is detailed and well presented.

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