Functional Analysis in Normed Spaces

additive and homogeneous adjoint B-space barrelled space bornological bounded set bounded variation closure compact consider continuous function COROLLARY corresponding D₁ defined definition denote dense denumerable domain element equation exists fact finite finite-dimensional follows Fourier function of bounded functional f fundamental system given H₁ Hence Hilbert space implies inequality interval inverse isometric kernel lemma linear functional linear operation linear set linear topological space linearly locally convex space matrix metric space necessary and sufficient neighbourhood of zero normed space obtained obviously orthogonal orthonormal system polynomial Proof properties reflexive Remark satisfies the conditions scalar product self-adjoint operator space H space X space Xn sphere subspace sufficient condition summable symmetric system of neighbourhoods topological space topology U₁ Un(x V₁ weak convergence weakly whence write x₁ Xn+1 Y₁