The course Methods of Applied Mathematics I (M383C) was taught by Dr. Irene Gamba.
Here’s the solution to problem 45 in Chapter 2 of the class textbook. The first part is an easy application of Arzela-Ascoli, while the second part is more involved. From what I understand, there is no norm defined on . The obvious supremum norm does not work with this space, for continuous functions such as polynomials would have infinite norm. So instead, I took the phrase “converge in ” to mean converge with respect to the topology generated by uniform convergence on compact subsets, i.e. the topology generated by the semi-norms
for each compact set .
I’m including a proposition here for my own sanity. When working with operators between Hilbert spaces, often the symbol is overloaded; In some contexts it stands for the adjoint map of , and in other contexts it stands for the dual map of . The following proposition shows why this choice of notation is not entirely un-justified.
Proposition: Let be a bounded linear operator between Hilbert spaces. There exists a unique mapping which satisfies
for all and . Moreover, this operator is induced by the dual map and the Riesz map.
Proof: There are two parts to the proof: uniqueness and existence. To show uniqueness, assume that there exist two operators that satisfy the condition. Then for every and ,
and since this occurs for all ,
This relation holds for all , so and we have uniqueness.
As for existence, let and be the Riesz maps for and . Let be the dual map defined by . Finally, set to be . We show that has the adjoint property.
Let and . First, we calculate
and hence evaluating at ,
Since and were arbitrary, this shows existence.