# Homework 1
### Updated until Sep 5; due Sep 12 in class.
$\renewcommand{\R}{\mathbb{R}}$
1. Suppose $f_n:\R\rightarrow \R, n\ge 1$ is a sequence of nonnegative functions such that
(1) Each $f_n$ is nonincreasing on $\R$, i.e. $f_n(x)\le f_n(y)$ whenever $x\ge y$
(2) $f_n\uparrow f$, i.e., $f_{n+1}(x)\ge f_n(x)$ for every $x\in \R$, converging pointwise as $n\rightarrow\infty$ to $f(x)$.
(3) $\int_0^\infty f_n(x)dx \le C$ for all $n$, for some constant $C$.
Verify that $$ \lim_{n\rightarrow \infty}\int_0^\infty f_n(x)dx = \int_0^\infty f(x)dx,$$ where the integrals are improper Riemann integrals.
2. Use the monotone convergence theorem to prove *Fatou's Lemma*: If $f_n$ is a sequence of nonnegative functions on $\R$ (not necessarily convergent) then
$$\int \lim\inf_{n\rightarrow \infty} f_n(x) dx \le \lim\inf_{n\rightarrow\infty} \int f_n(x)dx.$$
Give an example showing that the nonnegativity hypothesis cannot be removed.
3. Use Fatou's Lemma to prove the Dominated Convergence Theorem (Theorem I.11 of Reed-Simon). (hint: reduce to the nonnegative case first, and show that the necessary $\lim\inf = \lim\sup$ by considering a carefully chosen nonnegative auxiliary function.)
4. (optional) Prove that $L^1$ equipped with the $\|\cdot\|_1$ norm is a complete metric space (Riesz-Fisher Theorem, I.12 of Reed-Simon). The proof is actually given in Reed-Simon so you can either read it or try to do it yourself (I recommend the latter).
5. Prove that the inner product $(x,y)$ in a Hilbert space is jointly continuous in $x,y$ (Richtmyer 1.3.1)
6. Prove that the space $\ell^2$ is complete (Richtmyer 1.4.1-2)
7. Prove that if a Hilbert space has a countable basis then it is separable.
8. Richtmyer 1.7.1-1.7.6.
9. Prove that a linear functional on a hilbert space is bounded if and only if it is continuous.
10. (optional) Reed and Simon II.2.
11. Prove that every inner product space satisfies the *parallelogram identity*:
$$ \|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2.$$
Use this to show that $L^1(\R)$ is *not* a Hilbert space (i.e., its norm does not come from an inner product).

The official schedule is posted at https://www.jointmathematicsmeetings.org/meetings/national/jmm2023/2270_program_ss94.html#title The Zoom link for virtual attendance/virtual talks is: https://berkeley.zoom.us/j/94982941411?pwd=RWRXeVdoZUNVTmdqK1FIMksrZFllZz09 Friday, January 6 (101 Hynes Convention Center) 1-2pm: Matt Colbrook, The foundations of infinite-dimensional spectral computations (Zoom) 2-3pm: JosuĂ© Tonelli-Cueto, Condition-based Low-Degree Approximation of Real Polynomial Systems

1/3/2023Lecture 1: Propositional Logic A proposition is a declarative sentence that is either true or false but not both. A proposition has a truth value which is either $T$ or $F$. A letter used to denote a proposition is called a propositional variable. If $p$ is a proposition, the negation of $p$, denoted $\lnot p$, is the proposition "it is not the case that $p$". The truth value of $\lnot p$ is the opposite of that of $p$. If $p$ and $q$ are propositions, the conjunction of $p$ and $q$ (denoted $p\land q$) is the proposition "p and q." Its truth value is $T$ when both $p$ and $q$ are $T$, and it is $F$ otherwise. If $p$ and $q$ are propositions,he disjunction of $p$ and $q$ is the proposition "p or q". Its truth value is $T$ when at least one of $p$ or $q$ is $T$, and $F$ if both $p$ and $q$ are $F$. If $p$ and $q$ are propositions, the conditional $p\rightarrow q$ is the proposition "if $p$ then $q$". Its truth values are given by the following truth table (which lists its truth values in terms of those of $p$ and $q$): [see Table 5 in Rosen] The converse of $p\rightarrow q$ is $q\rightarrow p$. A compound proposition consists of logical operations (the four listed above) applied to propositions or propositional variables, possibly with parentheses. The truth value of a compound proposition can be mechanically determined given the truth values of its constituents.

12/12/2022Lecture 14 Basic Terminology, Handshaking Theorem A graph is a pair $G=(V,E)$ where $V$ is a finite set of vertices and $E$ is a finite multiset of $2-$element subsets of $V$, called edges. If $E$ has repeated elements $G$ is called a multigraph, otherwise it is called a simple graph. We will not consider directed graphs or graphs with loops (edges from a vertex to itself). Two vertices $x,y\in V$ are adjacent if ${x,y}\in E$. An edge $e={x,y}$ is said to be incident with $x$ and $y$. The degree of a vertex $x$ is the number of edges incident with it. Theorem. In any graph $G=(V,E)$, $$ \sum_{v\in V} deg(v) = 2|E|.$$ Proof. Let $I={(e,v):e\in E,v\in V,\textrm{$e$ is incident with $v$ in $G$}}$ be the set of edge-vertex incidences in $G$. We will count the number of incidences in two ways: (i) Observe that every edge participates in exactly two incidences, and no incidence participates in more than one edge. Thus, the total number of incidences is $|I|=2|E|$.

12/12/2022Prove that every simple graph with $n$ vertices and $k$ edges has at least $n − k$ connected components (hint: induction, or contradiction). Let $p\ge 3$ be a prime. Consider the graph $G=(V,E)$ with $V={0,1,2,\ldots, (p-1)}$ and $$E={{x,y}:x-y\equiv 2(mod p)\lor x-y\equiv -2 (mod p)}$$ Show that $G$ is connected. Does $G$ have an Euler circuit? Prove your answer. Let $n\ge 1$ be an integer and let $k\le n/2$. Consider the graph $G=(V,E)$ with $V={S\subseteq {1,2,\ldots,n}: |S|=k}$ and $$E = {{S,T}:|(S-T)\cup (T-S)|=1}.$$ What are the degrees of the vertices in $G$? Is $G$ connected? For which values of $n$ and $k$ does $G$ have an Eulerian circuit? For which values of $n$ and $k$ is $G$ $2-$colorable? Prove your answers.

11/8/2022
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