Chapter 1. Nonintegral dimensions and indices in analysis and geometry Although th e greater portio n o f thes e lectures will be concerne d wit h strictl y algebrai c questions, they ar e best motivate d b y considerin g certai n problem s that aris e i n analysis and geometry. W e shall attempt t o sketc h thi s background withou t goin g into details . Thi s ma- terial i s not neede d i n subsequent chapters,althoug h w e shal l return t o i t i n Chapte r 10 . Nonintegral dimension s for vecto r space s first explicitly aros e in the theor y o f Hilber t spaces. I f H is a separable infinit e dimensiona l Hilber t space , it ha s both discret e an d con - tinuous vector bases. T o explain this , let H = L 2 (T, dz) wher e (T, dz) denote s the uni t cir - cle with uni t Lebesgu e measure. W e may us e the ma p t —• e11 t o identif y (fO , 2n] *, dt\2n) with (T , dz), wher e [0 , 2n] * is [0 , 2TT] wit h 0 and 2n identified . Th e function s e n {t) = eint (n G Z) form a n orthonormal Hilber t basi s for H, i.e., for each/ G H, we have the L 2 - convergent sum s /(*) = ZAn)e„(t) (/(» ) = ^ f 2 o " AO^ W dt). This corresponds to th e Hilber t spac e direc t su m decomposition H e z // n , H n = Ce n . Given any subse t B C Z, we have a corresponding close d subspac e H B= Y,® H n = {f&H: ?{n) = Ofox n&B). The Hilber t spac e dimensio n o f H B i s given by di m HB = car d B an d w e have that H B an d Hc ar e isometric i f an d onl y i f di m HB = di m H c . On the othe r hand , if we let H = L 2 (R, dt), wher e (R , dt) i s the rea l line wit h Le - besgue measure (recall that all separable infinite dimensiona l Hilbert spaces are isometric), Four- ier integrals provid e a natural continuou s basi s for H, Lettin g e a (t) = e , a E R (w e nota - tionally distinguis h R from it s Pontrjagin dual) , we have /(o = XI#«)*«.(')da(fa ) = h /"- m ^ d °)- Thus in som e sense the se t { e a } a e R shoul d b e regarded a s a basis fo r H, eve n thoug h th e functions e a ar e not element s o f H. W e have a corresponding direc t integra l decompositio n H f^H a daf H a Q£tCL where the H a ar e no t actuall y subspace s of H. Give n a measur- able subse t 5 £ R ) W e ma y le t H B = SBHa da = {fEH: f(a) = 0 for a g B} . Althoug h 3 http://dx.doi.org/10.1090/cbms/046/03
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