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.