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Putnam
1966 Putnam
A5
A5
Part of
1966 Putnam
Problems
(1)
Putnam 1966 A5
Source:
4/6/2022
Let
C
C
C
denote the family of continuous functions on the real axis. Let
T
T
T
be a mapping of
C
C
C
into
C
C
C
which has the following properties:1.
T
T
T
is linear, i.e.
T
(
c
1
ψ
1
+
c
2
ψ
2
)
=
c
1
T
ψ
1
+
c
2
T
ψ
2
T(c_1\psi _1+c_2\psi _2)= c_1T\psi _1+c_2T\psi _2
T
(
c
1
ψ
1
+
c
2
ψ
2
)
=
c
1
T
ψ
1
+
c
2
T
ψ
2
for
c
1
c_1
c
1
and
c
2
c_2
c
2
real and
ψ
1
\psi_1
ψ
1
and
ψ
2
\psi_2
ψ
2
in
C
C
C
.2.
T
T
T
is local, i.e. if
ψ
1
≡
ψ
2
\psi_1 \equiv \psi_2
ψ
1
≡
ψ
2
in some interval
I
I
I
then also
T
ψ
1
≡
T
ψ
2
T\psi_1 \equiv T\psi_2
T
ψ
1
≡
T
ψ
2
holds in
I
I
I
.Show that
T
T
T
must necessarily be of the form
T
ψ
(
x
)
=
f
(
x
)
ψ
(
x
)
T\psi(x)=f(x)\psi(x)
T
ψ
(
x
)
=
f
(
x
)
ψ
(
x
)
where
f
(
x
)
f(x)
f
(
x
)
is a suitable continuous function.
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