Functional Analysis

absolutely convex apply approximate solutions arbitrary assume B-space ball bounded Cauchy sequence characteristic value closed coefficients compact operator compactum complete consider continuous functions continuous inverse continuous linear functional continuous linear operator convergence convex COROLLARY corresponding countable D₁ defined denote derivative differentiable eigenvalue equation exists fact finite finite-dimensional following theorem formula H₁ Hence Hilbert space inequality integral operator isomorphic kernel Lemma linear manifold matrix measure metric space n₁ necessary and sufficient neighbourhood of zero Newton's method non-empty normed space obtain orthogonal P(xo P₁ polynomial Proof properties prove relatively compact REMARK right-hand side satisfies condition self-adjoint operator simplexes steepest descent subset subspace sufficient condition Suppose t₁ theory topology U₁ unique vector space verify weak weak topology weakly write x₁ y₁