2. Definition of a distribution 7
Then f(x) is a linear functional. If ϕm(x) ϕ in D, then supp ϕm
BR0 for some R0 0 and maxx∈Rn |ϕm(x)−ϕ(x)| 0 as m ∞. Therefore
|f(ϕm) f(ϕ)|
BR0
|f(x)||ϕm(x) ϕ(x)|dx
max
x∈Rn
|ϕm(x) ϕ(x)|
BR0
|f(x)|dx 0
as m ∞.
b) Let f(x) be as in a). Denote by fk(ϕ) the following linear functional:
fk(ϕ) =
Rn
f(x)
∂kϕ(x)
∂xk
dx.
Then fk(ϕ) is also a distribution since ϕm ϕ in D implies that
maxx∈Rn |
∂kϕ(x)
∂xk

∂kϕm(x)
∂xk
| 0.
c) Delta-function is a distribution defined by the formula
δ(ϕ) = ϕ(0).
It is clear that δ(ϕ) is a linear continuous functional since ϕm(x) ϕ(x)
implies that ϕm(0) ϕ(0).
d) For n = 2 we introduce polar coordinates
x1 = r cos θ, x2 = r sin θ, r = x1
2
+
x22
and define a linear functional f(ϕ) by the formula
f(ϕ) =

0
ϕ(cos θ, sin θ) dθ,
i.e., ϕ(x1,x2) is integrated over the unit circle. It is clear that f(ϕ) is a
distribution.
2.2. Regular functionals.
We call the distribution defined by formula (2.1) the regular functional cor-
responding to the locally integrable function f(x). Two distributions f1(ϕ)
and f2(ϕ) are called equal if f1(ϕ) = f2(ϕ) for all ϕ D.
Proposition 2.1. Let f1(ϕ) and f2(ϕ) be two regular functionals corre-
sponding to f1(x) and f2(x). Then f1(ϕ) = f2(ϕ), ∀ϕ D, iff f1(x) = f2(x)
almost everywhere.
Proof: We have
f1(ϕ) f2(ϕ) =
Rn
(f1(x) f2(x))ϕ(x)dx = 0, ∀ϕ D.
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