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
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
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