Fall 2018 Practice Midterm 2.1 #real_analysis/qual/work
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Carefully state Tonelli’s theorem for a nonnegative function \(F(x, t)\) on \({\mathbb{R}}^n\times{\mathbb{R}}\).
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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
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Let \(f, g\in L^1({\mathbb{R}}^n)\) and give a definition of \(f\ast g\).
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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*}
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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 \(H\).
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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*}
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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.
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Take \(\left\{{a_n}\right\} \in \ell^2\), then note that \(\sum {\left\lvert {a_n} \right\rvert}^2 < \infty \implies\) the tails vanish.
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Define \(x \coloneqq\displaystyle\lim_{N\to\infty} S_N\) where \(S_N = \sum_{k=1}^N a_k u_k\)
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\(\left\{{S_N}\right\}\) is Cauchy and \(H\) is complete, so \(x\in H\).
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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.
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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.
` \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
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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*}
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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 \(f+g\).
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Let \(X = [0, 1] \subset {\mathbb{R}}\).
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Give a definition of the Banach space \(L^\infty(X)\) of essentially bounded functions of \(X\).
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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
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\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.
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Give the definition of what it means for a map \(L:X\to {\mathbb{C}}\) to be a linear functional.
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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.