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 — 2®

e z

//

n

, H

n

= Ce

n

.

Given any subse t B C Z, we have a corresponding close d subspac e

HB= 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 adaf 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

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http://dx.doi.org/10.1090/cbms/046/03