Methods of Applied Mathematics I, Fall 2018

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 \mathcal{C}^0(\mathbb{R}). 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 \mathcal{C}^0( \mathbb{R})” to mean converge with respect to the topology generated by uniform convergence on compact subsets, i.e. the topology generated by the semi-norms

\Vert f \Vert_{\mathcal{C}^0(K)} : = \sup_{x \in K} |f(x)|

for each compact set K \subset \mathbb{R}.


I’m including a proposition here for my own sanity. When working with operators between Hilbert spaces, often the symbol T^* is overloaded; In some contexts it stands for the adjoint map of T, and in other contexts it stands for the dual map of T. The following proposition shows why this choice of notation is not entirely un-justified.

Proposition: Let T: H_1 \rightarrow H_2 be a bounded linear operator between Hilbert spaces. There exists a unique mapping T^\dagger : H_2 \rightarrow H_1 which satisfies

(x,T^\dagger y)_{H_1} = (Tx, y)_{H_2}

for all x \in H_1 and y \in H_2. Moreover, this operator is induced by the dual map T^*: H_2^* \rightarrow H_1^* and the Riesz map.

Proof: There are two parts to the proof: uniqueness and existence. To show uniqueness, assume that there exist two operators T_1^\dagger , T_2^\dagger : H_2 \rightarrow H_1 that satisfy the condition. Then for every x \in H_1 and y \in H_2,

(Tx,y)_{H_2} = (x ,T_1^\dagger y )_{H_1} = (x,T_2^\dagger y)_{H_1}

Hence

0= (x ,T_1^\dagger y )_{H_1} - (x,T_2^\dagger y)_{H_1} = (x, T_1^\dagger y - T_2^\dagger y)_{H_1}

and since this occurs for all x \in H_1,

T_1^\dagger y = T_2^\dagger y.

This relation holds for all y \in H_2, so T_1^\dagger = T_2^\dagger and we have uniqueness.

As for existence, let R_1: H_1 \rightarrow H_1^* and R_2 : H_2 \rightarrow H_2^* be the Riesz maps for H_1 and H_2. Let T^* : H_2^*\rightarrow H_1^* be the dual map defined by T^*(f) = f \circ T. Finally, set S:H_2 \rightarrow H_1 to be S : = R_1^{-1} T^* R_2. We show that S has the adjoint property.

Let x\in H_1 and y \in H_2. First, we calculate

R_1(Sy) = R_1(R_1^{-1} T^* R_2 y )= T^* R_2 y =(R_2y)T

and hence evaluating at x,

(x, Sy)_{H_1} = (Tx , y)_{H_2}.

Since x \in H_1 and y \in H_2 were arbitrary, this shows existence. \square


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