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adjoint arbitrary assume axioms Banach space basis bounded linear functional bounded linear transformation called Cauchy sequence compact complete computation concept conjugate consider converges convex set corollary countable definition denote determined diagonal differential equation eigenvalues element equivalent Example Exercise exists fact Figure finite finite-dimensional follows function spaces given Hahn-Banach theorem hence Hilbert space homogeneous hyperplane implies inequality infinite initial condition inner product input integral interval inverse isometric Lemma linear dynamic system linear operator linear subspace linear transformation linearly independent mapping metric space Minkowski functional nonsingular normal normed linear space notation null space one-to-one open sets optimal control orthogonal polynomials preimage problem Proof properties prove reader real numbers reflexive Remark result rotund satisfies scalar Section Similarly solution space H subset system of Eq THEOREM F transition matrix tuplet u,Fu unique unit ball values variables vector verify zero