Number Theory: An Introduction to Mathematics

11 Further Remarks 75

Analmost periodic function, in the sense of Bohr (1925), is a functionf:R→C
which can be uniformly approximated onRbygeneralized trigonometric polynomials

∑m

j= 1

cjeiλjt,

wherecj∈Candλj∈R(j= 1 ,...,m). For any almost periodic functionsf,g,the
limit

〈f,g〉=lim

T→∞

( 1 / 2 T)

∫T

−T

f(t)g(t)dt

exists. The setBof all almost periodic functions acquires in this way the structure of
an inner product space. The setE={eiλt:λ∈R}is an uncountable orthonormal set
and Parseval’s equality holds for this set.
A finite-dimensional inner product space isnecessarily complete as a metric space,
i.e., every fundamental sequence converges. However, an infinite-dimensional inner
product space need not be complete, asC(I)already illustrates. An inner product
space which is complete is said to be aHilbert space.
The case considered by Hilbert (1906) was the vector space^2 of all infinite
sequencesx=(ξ 1 ,ξ 2 ,...)of complex numbers such that

∑

k≥ 1 |ξk|

(^2) <∞, with
〈x,y〉=

∑

k≥ 1

ξkη ̄k.

Another example is the vector spaceL^2 (I),whereI =[0,1], of all (equivalence
classes of) Lebesgue measurable functionsf :I →Csuch that

∫ 1

0 |f(t)

(^2) dt<∞,
with
〈f,g〉=

∫ 1

0

f(t)g(t)dt.

With any f ∈L^2 (I)we can associate a sequencefˆ∈^2 , consisting of the in-
ner products〈f,en〉,whereen(t)=e^2 πint(n∈Z),insomefixedorder.Themap
F:L^2 (I)→^2 thus defined is linear and, by Parseval’s equality,

‖Ff‖=‖f‖.

In factFis anisometrysince, by thetheorem of Riesz–Fischer(1907), it is bijective.

11 FurtherRemarks

A vast fund of information about numbers in different cultures is contained in
Menninger [52]. A good popular book is Dantzig [18].
The algebra of sets was created by Boole (1847), who used the symbols+and·
instead of∪and∩, as is now customary. His ideas were further developed, with appli-
cations to logic and probability theory, in Boole [10]. A simple system of axioms for