8 I. Theory of Distributions
Taking real-valued ϕ D and separating the real and the imaginary part of
the integral, we can assume without loss of generality that f1(x) f2(x) is
real-valued. Let
sgn(f1(x) f2(x)) =



1 if f1(x) f2(x) 0,
0 if f1(x) f2(x) = 0,
−1 if f1(x) f2(x) 0.
Then sgn(f1 f2) is a bounded measurable function.
Fix an arbitrary R 0. It is known from Lebesgue measure theory (see,
for example, [R]) that there exists a sequence of step-functions {sm(x)}
such that sm(x) sgn(f1 f2) almost everywhere in BR as m and
|sm(x)| 1. For any step-function sm(x), there exists a sequence of continu-
ous functions {cp(x)} with supports in BR such that cp(x) sm(x) almost
everywhere (a.e.) in BR as p and |cp(x)| 1. Using these two facts
and the Proposition 1.3 we can find a sequence ϕm(x) of C0
∞-functions
with
supports in BR such that ϕm(x) sgn(f1 −f2) a.e. in BR and |ϕm(x)| 1.
Then the Lebesgue convergence theorem implies that
(f1(x) f2(x))ϕm(x)dx
BR
|f1(x) f2(x)|dx as m ∞.
Thus
BR
|f1(x) f2(x)|dx = 0. Therefore f1(x) = f2(x) a.e. in BR. Since
R is arbitrary, we get f1(x) = f2(x) a.e. in
Rn.
The distributions form a linear space since a linear combination α1f1 +
α2f2 of distributions is also a distribution:
(α1f1 + α2f2)(ϕ) = α1f1(ϕ) + α2f2(ϕ).
The linear space of distributions is denoted by D .
2.3. Distributions in a domain.
Let Ω
Rn
be an arbitrary domain. The space D(Ω) is the space C0
∞(Ω)
with the following notion of convergence: ϕn(x) C0
∞(Ω)
converges to
ϕ(x) C0
∞(Ω)
in D(Ω) if there exists a compact subdomain B Ω such
that supp ϕn B for all n 1 and
∂kϕn(x)
∂xk

∂kϕ
∂xk
uniformly on B for all
0 |k| +∞.
The space of distribution D (Ω) is the space of all linear continuous
functionals on D(Ω).
Let U
Rn
be an open set and let f D
(Rn).
We shall define the
restriction f|U of f to U as f|U (ϕ) = f(ϕ) for all ϕ D(U). It is obvious
that f|U D (U).
Example 2.1 (A nonextendible distribution). Let f(x1) =
ex1
2
. Consider
the regular functional in D (0, +∞) defined by f(ϕ) =

0
e
1
x2
1
ϕ(x1)dx1,
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