Midterm Exam 2 (November 2018)

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 $\phi\in L^1({\mathbf{R}}^n)$ 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 $g\in L^\infty([0, 1])$.

• Prove $$\begin{align*} {\left\lVert {g} \right\rVert}_{L^p([0, 1])} \overset{p\to\infty}\to {\left\lVert {g} \right\rVert}_{L^\infty([0, 1])} .\end{align*}$$

• Prove that the map $$\begin{align*} \Lambda_g: L^1([0, 1]) &\to {\mathbf{C}}\\ f &\mapsto \int_0^1 fg \end{align*}$$ 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}