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Basic Properties of Hilbert Spaces
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absolutely continuous adjoint Borel sets bounded linear bounded operator Cauchy sequence clearly closed linear compact continuous function continuous linear functional converges strongly converges weakly convex set Corollary corresponding countably additive cylinder set deﬁned deﬁnes a linear dense domain du(t eigenvalue element example ﬁeld ﬁnite dimensional ﬁrst ﬁxed function f given goes to zero Hilbert space Hilbert—Schmidt implies inner product interior point kernel Lebesgue measurable Lemma Let 9 Let f linear bounded transformation linear operator linear space linear subspace Moreover n-linear form nonnegative deﬁnite nonzero norm nuclear operator mapping 9 optimal control orthonormal basis probability measure projection operator PROOF random variable readily veriﬁed resolvent set satisﬁes Schwarz inequality Section semigroup separable Hilbert space set in 9 solution space 9 strongly continuous subset Suppose Theorem theory unique vector Volterra operator weak convergence