Functional analysis

absolutely convex apply arbitrary assume B-space ball Banach bounded called Cauchy sequence characteristic value choose clearly closed coefficients coincides compact operator compactum consider continuous functions continuous inverse continuous linear functional continuous linear operator convergence convex set Corollary corresponding countable defined definition denote dense derivative differentiable eigenvalue element equation equivalent exists fact finite number finite-dimensional following theorem formula Hence Hilbert space implies inequality integral operator isomorphic kernel Lemma linear manifold matrix measurable functions measure space metric space Moreover necessary and sufficient neighbourhood of zero Newton's method non-empty normed space obtain obviously open set orthogonal polynomial properties relatively compact Remark right-hand side satisfies condition self-adjoint operator simplexes steepest descent subset subspace sufficient condition summable Suppose theory topological space unique vector space verify weak topology weakly write