THE RISE OF FUNCTIONS
13
This last proposition was a very technical statement from working
mathematics having no unusual features at all and yet it had the following
momentous consequences. After proving the proposition, Cantor asked
himself whether, in the hypothesis, the convergence to 0 has to indeed
be known for all points x of the interval, or whether there might not be in
the interval an “exceptional” set £ for the points of which nothing is stipu-
lated. Cantor found, successively, that the following sets are exceptional:
(a set consisting of) a single point; a finite number of points; a set having
a single accumulation point (and Cantor defines an accumulation point
for the occasion); a set having finitely many accumulation points; a set E
whose set of accumulation points E' has a single accumulation point or
finitely many accumulation points; etc.
This led Cantor to the conception of a transfinite ordinal number, and
thence to the conception and theory of pointsets and also of general aggre-
gates; and the world of thought, of any thought, has not been the same
since.
Orthogonal Systems. The trigonometric representation (5) of /(x) is an
expansion of /(x) into an orthogonal system, and it is remarkable that
mathematicians did not realize this feature of trigonometric functions for
a century or longer. The 19th century discovered many other complete
orthogonal systems among various families of special functions, functions
of Bessel, Lamé, Lagrange, Laguerre, Hermite, Jacobi, Heine, Gegenbauer,
and others. It was discovered for each such system separately that general
functions can be expanded in terms of it, and it was even known, after a
fashion, that every system was a complete set of eigenfunctions associated
with an elliptic differential equation. But somehow in the 19th century these
separate facts were not properly linked up; the accumulated knowledge was
broad and eclectic rather than compact and systematic and did not contribute
to the general theory of functions of a real variable. Riemann, for instance,
took no notice of the special functions of this kind (except for hypergeomet-
ric functions, which, however, were holomorphic functions for him), and
there does not seem to be a single “Riemann formula” about them.
But after the turn of the century the study of orthogonality suddenly
became a serious mathematical occupation. Its achievements were spear-
headed by the Riesz-Fischer theorem, which was a great triumph of the
newly conceived Lebesgue integral, and above all by Hilbert’s spectral
decomposition of a bounded self-adjoint operator in Hilbert space. This in
turn led to Lp-spaces, Banach spaces, and functional analysis.
Early functional analysis in the 19th century distinguished between