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2 The HahnBanach Theorem
3 The Separation Theorems
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Amer apply assume Banach space best approximation bounded characterization Chebyshev subspace condition constraints continuous functions convergence convex cone convex function convex program convex set convex subset Corollary CR(ft defined Definition dual space Duality E-property E-space element epi f equation example Exercise existence ext U(X extreme point f e Conv finite dimensional follows formula given Haar subspace hence Hilbert space implies inequality inner product space inverse Krein-Milman Theorem Lagrange multiplier Lagrange multiplier vector Lemma Let f linear functionals linear space linear subspace linearly independent mapping Math metric projections n-dimensional open set optimization ordinary convex program particular PM(x polynomial problem Proof proximinal pseudoinverse real lcs reflexive reflexive space resp result scalars sequence smooth solvable subdifferential subgradient Suppose Theorem theory topology uniformly rotund unique x-xQ