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) =

2π

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.