Functional Analysis in Normed Spaces

approximate arbitrary assume B-space barrelled space bornological bornological space bounded set bounded variation characteristic value closed set compact completely continuous operation consider continuous function convex set COROLLARY corresponding D₁ defined denote denumerable eigenvalue element equation exists fact finite finite-dimensional follows function of bounded functional f fundamental system given Hence Hilbert space implies inequality integral kernel lemma linear functional linear operation linear set linear topological space linearly locally convex space mapping metric space n₁ necessary and sufficient neighbourhood of zero normed space notation obtain obviously orthogonal P₁ P₂ polynomial Proof properties Remark right-hand side satisfies the conditions self-adjoint sequence solution space Xn subspace sufficient condition symmetric system of neighbourhoods topological space topology U₁ unique weak convergence weakly whence write x₁ Xn+1 Y₁ Y₂