Practice Exam (November 2014)

Fall 2018 Practice Midterm 2.1 #real_analysis/qual/work

Carefully state Tonelli’s theorem for a nonnegative function $F(x, t)$ on ${\mathbb{R}}^n\times{\mathbb{R}}$ .
Let $f:{\mathbb{R}}^n\to [0, \infty]$ and define $\begin{align*} {\mathcal{A}}\coloneqq\left\{{(x, t) \in {\mathbb{R}}^n\times{\mathbb{R}}{~\mathrel{\Big\vert}~}0\leq t \leq f(x)}\right\} .\end{align*}$

Prove the validity of the following two statements:
- $f$ is Lebesgue measurable on ${\mathbb{R}}^{n} \iff {\mathcal{A}}$ is a Lebesgue measurable subset of ${\mathbb{R}}^{n+1}$ .
- If $f$ is Lebesgue measurable on ${\mathbb{R}}^n$ then $\begin{align*} m(\mathcal{A})=\int_{\mathbb{R}^{n}} f(x) d x=\int_{0}^{\infty} m\left(\left\{x \in \mathbb{R}^{n}{~\mathrel{\Big\vert}~}f(x) \geq t\right\}\right) d t .\end{align*}$

Fall 2018 Practice Midterm 2.2 #real_analysis/qual/work

Let $f, g\in L^1({\mathbb{R}}^n)$ and give a definition of $f\ast g$ .
Prove that if $f, g$ are integrable and bounded, then $\begin{align*} (f\ast g)(x) \overset{{\left\lvert {x} \right\rvert}\to\infty}\to 0 .\end{align*}$

In parts:
- Define the Fourier transform of an integrable function $f$ on ${\mathbb{R}}^n$ .
- Give an outline of the proof of the Fourier inversion formula.
- Give an example of a function $f\in L^1({\mathbb{R}}^n)$ such that $\widehat{f}$ is not in $L^1({\mathbb{R}}^n)$ .

Fall 2018 Practice Midterm 2.3 #real_analysis/qual/work

\label{hilbert_space_exam_question}

Let $\left\{{u_n}\right\}_{n=1}^\infty$ be an orthonormal sequence in a Hilbert space $H$ .

Let $x\in H$ and verify that $\begin{align*} \left\|x-\sum_{n=1}^{N}\left\langle x, u_{n}\right\rangle u_{n}\right\|_H^{2} = \|x\|_H^{2}-\sum_{n=1}^{N}\left|\left\langle x, u_{n}\right\rangle\right|^{2} .\end{align*}$ for any $N\in {\mathbb{N}}$ and deduce that $\begin{align*} \sum_{n=1}^{\infty}\left|\left\langle x, u_{n}\right\rangle\right|^{2} \leq\|x\|_H^{2} .\end{align*}$
Let $\left\{{a_n}\right\}_{n\in {\mathbb{N}}} \in \ell^2({\mathbb{N}})$ and prove that there exists an $x\in H$ such that $a_n = {\left\langle {x},~{u_n} \right\rangle}$ for all $n\in {\mathbb{N}}$ , and moreover $x$ may be chosen such that $\begin{align*} {\left\lVert {x} \right\rVert}_H = \qty{ \sum_{n\in {\mathbb{N}}} {\left\lvert {a_n} \right\rvert}^2}^{1\over 2} .\end{align*}$

Prove that if $\left\{{u_n}\right\}$ is complete, Bessel’s inequality becomes an equality.

solution (part b):

Take $\left\{{a_n}\right\} \in \ell^2$ , then note that $\sum {\left\lvert {a_n} \right\rvert}^2 < \infty \implies$ the tails vanish.
Define $x \coloneqq\displaystyle\lim_{N\to\infty} S_N$ where $S_N = \sum_{k=1}^N a_k u_k$
$\left\{{S_N}\right\}$ is Cauchy and $H$ is complete, so $x\in H$ .
By construction, $\begin{align*} {\left\langle {x},~{u_n} \right\rangle} = {\left\langle {\sum_k a_k u_k},~{u_n} \right\rangle} = \sum_k a_k {\left\langle {u_k},~{u_n} \right\rangle} = a_n \end{align*}$ since the $u_k$ are all orthogonal.
By Pythagoras since the $u_k$ are normal, $\begin{align*} {\left\lVert {x} \right\rVert}^2 = {\left\lVert {\sum_k a_k u_k} \right\rVert}^2 = \sum_k {\left\lVert {a_k u_k} \right\rVert}^2 = \sum_k {\left\lvert {a_k} \right\rvert}^2 .\end{align*}$

solution (part c):

Let $x$ and $u_n$ be arbitrary.

` \begin{align*} {\left\langle {x - \sum_^\infty {\left\langle {x},~{u_k} \right\rangle}u_k },~{u_n} \right\rangle} &= {\left\langle {x},~{u_n} \right\rangle}

{\left\langle {\sum_^\infty {\left\langle {x},~{u_k} \right\rangle}u_k },~{u_n} \right\rangle} \ &= {\left\langle {x},~{u_n} \right\rangle}

\sum_^\infty {\left\langle {{\left\langle {x},~{u_k} \right\rangle}u_k },~{u_n} \right\rangle} \ &= {\left\langle {x},~{u_n} \right\rangle}

\sum_^\infty {\left\langle {x},~{u_k} \right\rangle} {\left\langle {u_k },~{u_n} \right\rangle} \ &= {\left\langle {x},~{u_n} \right\rangle} - {\left\langle {x},~{u_n} \right\rangle} = 0 \ \implies x - \sum_^\infty {\left\langle {x},~{u_k} \right\rangle}u_k &= 0 \quad\text{by completeness} .\end{align*} `{=html}

So $\begin{align*} x = \sum_{k=1}^\infty {\left\langle {x},~{u_k} \right\rangle} u_k \implies {\left\lVert {x} \right\rVert}^2 = \sum_{k=1}^\infty {\left\lvert {{\left\langle {x},~{u_k} \right\rangle}} \right\rvert}^2. \hfill\blacksquare .\end{align*}$

Fall 2018 Practice Midterm 2.4 #real_analysis/qual/work

Prove Holder’s inequality: let $f\in L^p, g\in L^q$ with $p, q$ conjugate, and show that $\begin{align*} {\left\lVert {fg} \right\rVert}_{p} \leq {\left\lVert {f} \right\rVert}_{p} \cdot {\left\lVert {g} \right\rVert}_{q} .\end{align*}$
Prove Minkowski’s Inequality: $\begin{align*} 1\leq p < \infty \implies {\left\lVert {f+g} \right\rVert}_{p} \leq {\left\lVert {f} \right\rVert}_{p}+ {\left\lVert {g} \right\rVert}_{p} .\end{align*}$ Conclude that if $f, g\in L^p({\mathbb{R}}^n)$ then so is $f+g$ .

Let $X = [0, 1] \subset {\mathbb{R}}$ .
- Give a definition of the Banach space $L^\infty(X)$ of essentially bounded functions of $X$ .
- Let $f$ be non-negative and measurable on $X$ , prove that $\begin{align*} \int_X f(x)^p \,dx \overset{p\to\infty}\to \begin{dcases} \infty \quad\text{or} \\ m\qty{\left\{{f^{-1}(1)}\right\}} \end{dcases} ,\end{align*}$ and characterize the functions of each type

solution:

$\begin{align*} \int f^p &= \int_{x < 1} f^p + \int_{x=1}f^p + \int_{x > 1} f^p\\ &= \int_{x < 1} f^p + \int_{x=1}1 + \int_{x > 1} f^p \\ &= \int_{x < 1} f^p + m(\left\{{f = 1}\right\}) + \int_{x > 1} f^p \\ &\overset{p\to\infty}\to 0 + m(\left\{{f = 1}\right\}) + \begin{cases} 0 & m(\left\{{x\geq 1}\right\}) = 0 \\ \infty & m(\left\{{x\geq 1}\right\}) > 0. \end{cases} \end{align*}$

Fall 2018 Practice Midterm 2.5 #real_analysis/qual/work

Let $X$ be a normed vector space.

Give the definition of what it means for a map $L:X\to {\mathbb{C}}$ to be a linear functional.
Define what it means for $L$ to be bounded and show $L$ is bounded $\iff L$ is continuous.

Prove that $(X {}^{ \vee }, {\left\lVert {{-}} \right\rVert}_{^{\operatorname{op}}})$ is a Banach space.