# 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 $${\mathbf{R}}^n\times{\mathbf{R}}$$.

• Let $$f:{\mathbf{R}}^n\to [0, \infty]$$ and define \begin{align*} {\mathcal{A}}\coloneqq\left\{{(x, t) \in {\mathbf{R}}^n\times{\mathbf{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 $${\mathbf{R}}^{n} \iff {\mathcal{A}}$$ is a Lebesgue measurable subset of $${\mathbf{R}}^{n+1}$$.
• If $$f$$ is Lebesgue measurable on $${\mathbf{R}}^n$$ 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 $$f, g\in L^1({\mathbf{R}}^n)$$ 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 $${\mathbf{R}}^n$$.
• Give an outline of the proof of the Fourier inversion formula.
• Give an example of a function $$f\in L^1({\mathbf{R}}^n)$$ such that $$\widehat{f}$$ is not in $$L^1({\mathbf{R}}^n)$$.

## 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.

• 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*}

Let $$x$$ and $$u_n$$ be arbitrary.

## \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 $$f, g\in L^p({\mathbf{R}}^n)$$ then so is $$f+g$$.

• Let $$X = [0, 1] \subset {\mathbf{R}}$$.

• 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

\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 $$L:X\to {\mathbf{C}}$$ 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.
#real_analysis/qual/work