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 \({\left\lVert {f - P} \right\rVert}_{\infty} < {\varepsilon}\).