MathDB

Putnam 1966 A5

Source:

April 6, 2022

college contests

Problem Statement

Let

C

denote the family of continuous functions on the real axis. Let

T

be a mapping of

C

into

C

which has the following properties:
1.

T

is linear, i.e.

T(c_1\psi _1+c_2\psi _2)= c_1T\psi _1+c_2T\psi _2

for

c_1

and

c_2

real and

\psi_1

and

\psi_2

in

C

.
2.

T

is local, i.e. if

\psi_1 \equiv \psi_2

in some interval

I

then also

T\psi_1 \equiv T\psi_2

holds in

I

.
Show that

T

must necessarily be of the form

T\psi(x)=f(x)\psi(x)

where

f(x)

is a suitable continuous function.

Back to Problems View on AoPS