The 2020 exam

Section H.7 The 2020 exam

Links.

The 2020 exam was closed book. I did not write this exam. Below I report the point totals but also the ‘classification’ of each problem, usually as reported in the official solutions but sometimes with an additional comment of my own.

The term bookwork refers to a question which could be correctly answered by simply copying a relevant passage of text from the lecture notes. Several of the questions artificially restrict which results you are allowed to use in your arguments. This is something which will not happen on this year’s exam, see Subsection H.1.3 above.

Another small notational thing to be aware of is that in previous years we wrote \(L^2(a,b)\) rather than \(L^2([a,b])\text{.}\)

Up until last year, these notes used superficially different but mathematically equivalent definitions of “totally bounded”, “sequentially compact” and “compact”. For this reason, the official solution to Question 2 does not look exactly like Definition 4.7. The two definitions are equivalent by Lemma 4.8; for a discussion of why this change was made you can see entry for 2023/2024 in the beginning of the preface.

Exercises Annotated version of the 2020 exam

1. Question 1.

Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces.

[Bookwork, 6]
1. State what it means for \((X,d_X)\) to be complete.
2. State what it means for \(f \maps X \to Y\) to be an isometry and \((X,d_X)\) and \((Y,d_Y)\) to be isometric.
3. Define the concept of a completion of \((X,d_X)\text{.}\)
[Bookwork, 2] State a theorem regarding the existence of a completion of \((X,d_X)\) and a theorem regarding the relationship between any two completions of \((X,d_X)\text{.}\)
[Bookwork, 4] Suppose \((X,d_X)\) is a complete metric space and let \(Y \subseteq X\text{.}\) Prove that the subspace \((Y,d_Y)\) is complete if and only if \(Y\) is closed. You may use the following facts without proof: A subset \(S\) of a metric space \((X,d_X)\) is closed if and only if for every sequence \(S\) which converges in \(X\text{,}\) the limit belongs to \(S\text{;}\) Any convergent sequence in a metric space is Cauchy; Limits of convergent sequences in metric spaces are unique.
[Unseen, 4] Suppose that \((X,d_X)\) is complete and \((X,d_X)\) and \((Y,d_Y)\) are isometric. Prove that \((Y,d_Y)\) is complete.
[Unseen in 2019–2020, but seen this year, 4] Suppose \((X,d_X)\) is a complete metric space, let \(S \subseteq X\) and let \((\hat S, d_{\hat S})\) be any completion of the metric subspace \((S,d_S)\text{.}\) Let \(\overline S\) denote the closure of \(S\) in \(X\text{.}\) Show that the metric subspaces \((\overline S,d_{\overline S})\) and \((\hat S, d_{\hat S})\) are isometric. You may use any results from the course but state what you use.

2. Question 2.

Let \((X,d)\) be a metric space and \(C_\bdd(X)\) be the Banach space of bounded continuous functions \(f \maps X \to \R\) with the supremum norm.

[Bookwork, 6]
1. Define what it means for a set \(S \subseteq X\) to be totally bounded.
2. Define what it means for a set \(S \subseteq X\) to be compact in \(X\text{.}\)
3. Define what it means for a set \(S \subseteq X\) to be relatively compact in \(X\text{.}\)
4. Let \(F\) be a collection of maps \(f \maps X \to \R\text{.}\) Define what it means for \(F\) to be equicontinuous.
[Bookwork, 3] State the Arzelà–Ascoli Theorem without proof.
[Bookwork, 4] Suppose that a collection \(F \subset C_\bdd(X)\) is totally bounded. Prove directly that \(F\) is equicontinuous.
[Unseen, 4] Consider the Banach space \((C^0([a,b]), \n\blank_{C^0([a,b])})\) where \(\n f_{C^0([a,b])} = \sup_{x \in [a,b]} \abs{f(x)}\) and \([a,b]\) is equipped with the Euclidean metric. Suppose \(\seq fn\) is a sequence in \(C^0([a,b])\) with \(\sup_{n \in \N} \n{f_n}_{C^0([a,b])} \lt \infty\text{.}\) Let \(\seq Fn\) be defined by
\begin{equation*} F_n(x) = f_n(a) + \int_a^x f_n(t)\, dt. \end{equation*}
Prove that a subsequence of \(\seq Fn\) converges in \((C^0([a,b]), \n\blank_{C^0([a,b])})\text{.}\)
[Unseen, 3] Consider the metric space \(\R\) with the Euclidean metric. Using the function
\begin{equation*} f(x) = \begin{cases} x, \amp x \in [0,1] \\ 2-x, \amp x \in [1,2] \\ 0, \amp x \in \R \without [0,2] \end{cases}, \end{equation*}
or otherwise, find a sequence of functions \(\seq fn\text{,}\) with \(f_n \maps \R \to \R\) and \(f_n \in C_\bdd(\R)\) for every \(n \in \N\text{,}\) such that \(\set{f_n}{n \in \N}\) is bounded and equicontinuous but not relatively compact.

3. Question 3.

Let \((X,d)\) be a metric space and \(C_\bdd(X)\) be the Banach space of bounded continuous functions \(f \maps X \to \R\) with the supremum norm. Let \(S \subseteq C_\bdd(X)\text{.}\)

[Bookwork, 6]
1. State what it means for \(S\) to separate points.
2. Define the lattice operations \(\wedge \maps C_\bdd(X) \times C_\bdd(X) \to C_\bdd(X)\) and \(\vee \maps C_\bdd(X) \times C_\bdd(X) \to C_\bdd(X)\) and state what it means for \(S\) to be a lattice.
3. Suppose that \(S\) is a linear subspace. State what it means for \(S\) to be an algebra.
[Bookwork, 3] State the Stone–Weierstrass Theorem without proof.
[Was bookwork in 2019–2020, but might not be this year, 3] Let \(C_\bdd(X) \times C_\bdd(X)\) be the metric space with the product metric induced from \(C_\bdd(X)\text{.}\) Show that the operation \(C_\bdd(X) \times C_\bdd(X) \to C_\bdd(X) : (f,g) \mapsto fg\) is continuous.
[‘Essentially Bookwork’, 4] Consider the set \([0,1]^2 = [0,1] \times [0,1] \subset \R^2\) with the Euclidean metric. Let \(P([0,1]^2)\) denote the set of polynomials with real coefficients on \([0,1]^2\text{;}\) that is, \(p \in P([0,1]^2)\) if and only if we have \(p(x,y) = \sum_{j=0}^n \sum_{k=0}^n a_{j,k} x^j y^k\) for some \(n \in \N \cup \{0\}\text{,}\) where \((x,y) \in [0,1]^2\) and \(a_{j,k} \in \R\) for every \((j,k) \in \{0,1,2,\ldots,n\} \times \{0,1,2,\ldots,n\}\text{.}\) Show \(P([0,1]^2)\) is dense in \(C_\bdd([0,1]^2)\text{.}\)
[‘Seen and Unseen Exercise’, 4] Let \(P([a,b])\) be the space of polynomials with real coefficients on \([a,b]\text{;}\) that is, \(p \in P([a,b])\) if and only if \(p(x) = \sum_{k=0}^n a_k x^k\) for some \(n \in \N \cup \{0\}\text{,}\) \(a_k \in \R\) for \(k=0,\ldots,n\) and \(x \in [a,b]\text{.}\) Let \(P_\Q([a,b])\) be the subset of \(P([a,b])\) such that \(q \in P_\Q([a,b])\) if and only if \(q(x) = \sum_{k=0}^n b_k x^k\) for some \(n \in \N \cup \{0\}\text{,}\) \(b_k \in \Q\) for every \(k\) and \(x \in [a,b]\text{.}\)

Let \(C^0([a,b])\) be the real vector space of continuous functions on \([a,b]\) and consider the inner product \(\scp\blank\blank_{L^2([a,b])} \maps C^0([a,b]) \times C^0([a,b]) \to \R\) defined by

\begin{equation*} \scp fg_{L^2([a,b])} = \int_a^b f(x)g(x)\, dx \end{equation*}

for \(f,g \in C^0([a,b])\text{.}\) Show that the metric space \((C^0([a,b]),d_{L^2([a,b])})\) is separable where \(d_{L^2([a,b])}\) is the metric induced by the inner product \(\scp\blank\blank_{L^2([a,b])}\text{.}\)

You may use the fact that \(P_\Q([a,b])\) is countable without proof.

4. Question 4.

Let \((X,d)\) be a metric space.

[Bookwork, 6]
1. State what it means for \(X\) to be separable.
2. State what it means for a set \(Y\subseteq X\) to be dense.
3. State what it means for a set \(Y \subseteq X\) to be nowhere dense.
[Bookwork, 3] State the Baire Category Theorem without proof.
[Not specified in solutions, but Bookwork, 5] Prove the following statements. You may use the fact that a set \(S \subseteq X\) is dense in \(X\) if an only if \(\overline{S} = X\) and the Baire Category Theorem without proof.
1. A subset \(Y \subseteq X\) is nowhere dense if, and only if, the set \(X \without \overline Y\) is dense in \(X\text{.}\)
2. Suppose \((X,d)\) is complete and let \(\seq Yn\) be a sequence of nowhere dense subsets of \(X\text{.}\) Then \(X \without \cup_{n\in\N} Y_n\) is dense in \(X\text{.}\)
3. A non-empty, complete metric space is not a countable union of nowhere dense subsets.
[Unseen, 6] Let \((X,d)\) be a non-empty, complete metric space and \(F\) be a family of continuous functions \(f \maps X \to \R\) with the property that for every \(x \in X\) there exists \(M_x \in \R\) such that \(\abs{f(x)} \le M_x\) for every \(f \in F\text{.}\) In what follows you may use any result from the course but state what you use.
1. Let \(n \in \N\) and let \(A_n = \set{x \in X}{\abs{f(x)} \le n \text{ for every } f \in F}\text{.}\) Show \(A_n\) is closed. (Hint: First show the set \(A_{n,f}\) defined by \(A_{n,f} := \set{x \in X}{\abs{f(x)} \le n}\) is closed for every \(f \in F\text{.}\))
2. Show \(X = \cup_{n \in \N} A_n\text{.}\)
3. Show there exists \(x_0 \in X\text{,}\) \(r \gt 0\) and \(N \in \N\) such that for all \(x \in B_r(x_0)\) it holds that \(\abs{f(x)} \le N\) for every \(f \in F\text{.}\)

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