Linear Transformations in Hilbert Space and Their Applications to Analysis

according adjoint apply arbitrary arbitrary element assertion associated assumes belongs bounded Chapter characteristic value closed linear manifold coincides conclude condition consequence consider constant construct contains continuous converges corresponding defined definition denote described determined differential discussed domain easily element element f equal equation equivalent established everywhere dense evident exists expression extension fact find finite first fixed formation function function f given Hence Hilbert space holds identity implies independent indicated inequality infinite integral interval introduce inverse isometric Lemma limit manner matrix maximal means measure normal null obtain obvious operator orthogonal particular positive preceding problem projection proof properties prove range reduces reference relation respect result satisfied satisfy self-adjoint transformation sequence simple solution spectrum subset symmetric transformation takes Theorem theory trans transformation H unique vanishes virtue wished write zero