# Midterm Exam 2 (December 2014)

## Fall 2014 Midterm 1.1 #real_analysis/qual/work

Note: (a) is a repeat.

• Let $$\Lambda\in L^2(X) {}^{ \vee }$$.
• Show that $$M\coloneqq\left\{{f\in L^2(X) {~\mathrel{\Big\vert}~}\Lambda(f) = 0}\right\} \subseteq L^2(X)$$ is a closed subspace, and $$L^2(X) = M \oplus M\perp$$.
• Prove that there exists a unique $$g\in L^2(X)$$ such that $$\Lambda(f) = \int_X g \mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu$$.

## Fall 2014 Midterm 1.2 #real_analysis/qual/work

• In parts:
• Given a definition of $$L^\infty({\mathbb{R}}^n)$$.
• Verify that $${\left\lVert {{-}} \right\rVert}_\infty$$ defines a norm on $$L^\infty({\mathbb{R}}^n)$$.
• Carefully proved that $$(L^\infty({\mathbb{R}}^n), {\left\lVert {{-}} \right\rVert}_\infty)$$ is a Banach space.
• Prove that for any measurable $$f:{\mathbb{R}}^n \to {\mathbb{C}}$$, \begin{align*} L^1({\mathbb{R}}^n) \cap L^\infty({\mathbb{R}}^n) \subset L^2({\mathbb{R}}^n) {\quad \operatorname{and} \quad} {\left\lVert {f} \right\rVert}_2 \leq {\left\lVert {f} \right\rVert}_1^{1\over 2} \cdot {\left\lVert {f} \right\rVert}_\infty^{1\over 2} .\end{align*}

## Fall 2014 Midterm 1.3 #real_analysis/qual/work

• Prove that if $$f, g: {\mathbb{R}}^n\to {\mathbb{C}}$$ is both measurable then $$F(x, y) \coloneqq f(x)$$ and $$h(x, y)\coloneqq f(x-y) g(y)$$ is measurable on $${\mathbb{R}}^n\times{\mathbb{R}}^n$$.

• Show that if $$f\in L^1({\mathbb{R}}^n) \cap L^\infty({\mathbb{R}}^n)$$ and $$g\in L^1({\mathbb{R}}^n)$$, then $$f\ast g \in L^1({\mathbb{R}}^n) \cap L^\infty({\mathbb{R}}^n)$$ is well defined, and carefully show that it satisfies the following properties: \begin{align*} {\left\lVert {f\ast g} \right\rVert}_\infty &\leq {\left\lVert {g} \right\rVert}_1 {\left\lVert {f} \right\rVert}_\infty {\left\lVert {f\ast g} \right\rVert}_1 &\leq {\left\lVert {g} \right\rVert}_1 {\left\lVert {f} \right\rVert}_1 {\left\lVert {f\ast g} \right\rVert}_2 &\leq {\left\lVert {g} \right\rVert}_1 {\left\lVert {f} \right\rVert}_2 .\end{align*}

Hint: first show $${\left\lvert {f\ast g} \right\rvert}^2 \leq {\left\lVert {g} \right\rVert}_1 \qty{ {\left\lvert {f} \right\rvert}^2 \ast {\left\lvert {g} \right\rvert}}$$.

## Fall 2014 Midterm 1.4 #real_analysis/qual/work

Note: (a) is a repeat.

Let $$f: [0, 1]\to {\mathbb{R}}$$ be continuous, and prove the Weierstrass approximation theorem: for any $${\varepsilon}> 0$$ there exists a polynomial $$P$$ such that $${\left\lVert {f - P} \right\rVert}_{\infty} < {\varepsilon}$$.

#real_analysis/qual/work