Transcription of Chapter 8 Bounded Linear Operators on a Hilbert Space
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Chapter 8 Bounded Linear Operators on a Hilbert Space
Chapter8 BoundedLinearOperatorsona HilbertSpaceIn thischapterwe describe someimportant classesof boundedlinearoperatorsonHilbertspaces,in cludingprojections,unitaryoperators,ands elf-adjoint alsoprove theRieszrepresentationtheorem,which characterizestheboundedlinearfunctionals on a Hilbertspace,anddiscussweakconvergencein beginby describingsomealgebraicpropertiesof a linearspaceXsuch thateveryx2 Xcanbe writtenuniquelyasx=y+zwithy2 Mandz2N, thenwe say thatX=M Nis thedirect sumofMandN, andwe callNacomplementarysubspaceofMinX. Thedecompositionx=y+zwithy2 Mandz2 Nis uniqueif andonlyifM\N=f0g. A givensubspaceMhasmany ,ifX=R3andMis a planethroughtheorigin,thenany linethroughtheoriginthatdoes notlieinMis a ,andthedimensionof a complementarysubspaceis N, thenwe de netheprojectionP:X!
Thus, every bounded linear functional is given by the inner product with a xed vector. We have already seen that ’y(x) = hy;xi de nes a bounded linear functional on H for every y 2 H. To prove that there is a unique y in H associated with a given linear functional, suppose that ’y1 = ’y2. Then ’y1(y) = ’y2(y) when y = y1 y2,
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