Transcription of FUNCTIONAL ANALYSIS - Pitt
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FUNCTIONAL ANALYSIS - Pitt
FUNCTIONAL ANALYSISPIOTR HAJ and Hilbert spacesIn what followsKwill is a pair (X, ), whereXis a linear spaceoverKand :X [0, )is a function, called anorm, such that(1) x+y x + y for allx,y X;(2) x =| | x for allx Xand K;(3) x = 0 if and only ifx= x y x z + z y for allx,y,z X,d(x,y) = x y defines a metric in a normed space. In what follows normed paces will alwaysbe regarded as metric spaces with respect to the metricd. A normed spaceis called aBanach spaceif it is complete with respect to the a linear space overK(=RorC). Theinner product(scalar product) is a function , :X X Ksuch that(1) x,x 0;(2) x,x = 0 if and only ifx= 0;(3) x,y = x,y ;(4) x1+x2,y = x1,y + x2,y ;(5) x,y = y,x ,for allx,x1,x2,y Xand all an obvious corollary we obtain x,y1+y2 = x,y1 + x,y2 , x, y = x,y ,Date: February 12, HAJ LASZfor allx,y1,y2 Xand a space with an inner product we define x = x,x.]
FUNCTIONAL ANALYSIS 5 where U is unitary and Ris positive self-adjoint. The mapping Rcan be computed explicitly LLT = RUUTRT = R2, R= √ LLT. According to the spectral theorem there is an orthonormal basis v
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