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 ,

Hence

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.

Applied I – Homework 1

Applied I – Homework 2

Applied I – Homework 3

Applied I – Homework 4

Applied I – Homework 5

Midterm II Review

Midterm III Review

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