Fall 2014 Midterm 1.1 #real_analysis/qual/work
Note: (a) is a repeat.
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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 \(\Lambda(f) = \int_X g \overline{f}\).
Fall 2014 Midterm 1.2 #real_analysis/qual/work
- In parts:
- Given a definition of \(L^\infty({\mathbf{R}}^n)\).
- Verify that \({\left\lVert {{-}} \right\rVert}_\infty\) defines a norm on \(L^\infty({\mathbf{R}}^n)\).
- Carefully proved that \((L^\infty({\mathbf{R}}^n), {\left\lVert {{-}} \right\rVert}_\infty)\) is a Banach space.
- Prove that for any measurable \(f:{\mathbf{R}}^n \to {\mathbf{C}}\), \begin{align*} L^1({\mathbf{R}}^n) \cap L^\infty({\mathbf{R}}^n) \subset L^2({\mathbf{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
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Prove that if \(f, g: {\mathbf{R}}^n\to {\mathbf{C}}\) is both measurable then \(F(x, y) \coloneqq f(x)\) and \(h(x, y)\coloneqq f(x-y) g(y)\) is measurable on \({\mathbf{R}}^n\times{\mathbf{R}}^n\).
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Show that if \(f\in L^1({\mathbf{R}}^n) \cap L^\infty({\mathbf{R}}^n)\) and \(g\in L^1({\mathbf{R}}^n)\), then \(f\ast g \in L^1({\mathbf{R}}^n) \cap L^\infty({\mathbf{R}}^n)\) 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 \(f: [0, 1]\to {\mathbf{R}}\) be continuous, and prove the Weierstrass approximation theorem: for any \({\varepsilon}> 0\) there exists a polynomial \(P\) such that \({\left\lVert {f - P} \right\rVert}_{\infty} < {\varepsilon}\).