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Bounded Linear Operators and Bounded Invertibles

Let = c = C is a continuous function .
Let = sup : , for each f in Define T: by

(T ( ))(t) =
for each t , and For each f in .

a) Show that is a bounded linear operator on .
b) Compute , For each n in N, and compute .
c) Suppose that g . Show that the integral equation

(t) - = g (t) for each t in
has a solution f in

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Let = c = C is a continuous function .
Let = sup : , for each f in Define T: by
(T ( ))(t) =
for each t , and For each f in .

a) Show that T is a bounded linear operator on .

Proof:
(1) First we want to prove T is a linear operator, that is, T(f + g)=T(f) + T(g), for any
Actually, ...

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