Let K be a closed convex cone in a finite-dimensional linear space X and M a subspace

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Let K be a closed convex cone in a finite-dimensional linear space X and M a subspace with K ∩ M = {0}. Then there exists a linear functional f ∈ X* such that
f(x) > 0 for every x ∈ K\{0}
and
f(x) = 0 for every x ∈ M
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