Fall 2018 Midterm 2.1 #real_analysis/qual/work
Let \(f, g\in L^1([0, 1])\), define \(F(x) = \int_0^x f(y)\,dy\) and \(G(x) = \int_0^x g(y)\,dy\), and show \begin{align*} \int_0^1 F(x)g(x) \,dx = F(1)G(1) - \int_0^1 f(x) G(x) \, dx .\end{align*}
Fall 2018 Midterm 2.2 #real_analysis/qual/work
Let \(\phi\in L^1({\mathbb{R}}^n)\) such that \(\int \phi = 1\) and define \(\phi_t(x) = t^{-n}\phi(t^{-1}x)\). Show that if \(f\) is bounded and uniformly continuous then \(f\ast \phi_t \overset{t\to 0}\to f\) uniformly.
Fall 2018 Midterm 2.3 #real_analysis/qual/work
Let \(g\in L^\infty([0, 1])\).
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Prove \begin{align*} {\left\lVert {g} \right\rVert}_{L^p([0, 1])} \overset{p\to\infty}\to {\left\lVert {g} \right\rVert}_{L^\infty([0, 1])} .\end{align*}
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Prove that the map \begin{align*} \Lambda_g: L^1([0, 1]) &\to {\mathbb{C}}\\ f &\mapsto \int_0^1 fg \end{align*} defines an element of \(L^1([0, 1]) {}^{ \vee }\) with \({\left\lVert {\Lambda_g} \right\rVert}_{L^1([0, 1]) {}^{ \vee }}= {\left\lVert {g} \right\rVert}_{L^\infty([0, 1])}\).
Fall 2018 Midterm 2.4 #real_analysis/qual/work
See
\cref{hilbert_space_exam_question}