Department of Mathematics University of Utah Ph.D. Preliminary Examination in Analysis January 2020

Let \(\mathcal M\) be a \(\sigma \)-algebra of subsets of \(X\) and \(\mu \) a measure on \(\mathcal M\). Let \(A_1\subseteq A_2\subseteq \cdots \) be a sequence of sets in \(\mathcal M\). Let \[ A=\bigcup _{i=1}^{\infty } A_i. \] Prove that \(\lim _{i\to \infty }\mu (A_i)=\mu (A)\).

Now assume that \(\mu (X)<\infty \), and let \(A_1\supseteq A_2\supseteq \cdots \) be a sequence of sets in \(\mathcal M\). Let \[ A=\bigcap _{i=1}^{\infty } A_i. \] Prove that \(\lim _{i\to \infty }\mu (A_i)=\mu (A)\). Give an example where this fails if \(\mu (X)=\infty \).

Let \(f:[0,\infty )\to \mathbb R\) be defined by \(f(x)=e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral.

Let \(X\) be a normed space and \(X^*\) be the space of bounded linear functionals. Every \(x\in X\) defines a functional \(\hat {x}\) on \(X^*\) by \(\hat {x}(f)=f(x)\) for all \(f\in X^*\). Prove that \(\lVert \hat {x}\rVert =\lVert x\rVert \).

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Prove the following special case of the Baire category theorem, without directly using the general theorem: \(\mathbb R^2\) is not a countable union of lines.

State and prove Rouché’s theorem.

Give an example of a sequence of numbers \(a_1,\ldots \) so that there is no holomorphic function \(f:\mathbb D\to \mathbb C\) so that \(a_1=f'(0)\), \(a_2=f''(0)\), \(a_3=f'''(0)\), \(\ldots \). Justify your answer.

Show that if \(f\) is an entire function and \(f\) restricted to the unit disk is a bijection, that is, \(f|_{\mathbb D}:\mathbb D\to \mathbb D\) is a bijection, then \(f\) is a rotation. That is, there exists \(c\in \mathbb C\) with \(|c|=1\) so that \(f(z)=cz\) for all \(z\).

Using the methods of complex analysis compute \[ \int _0^\infty \frac {1}{x^4+16}\,dx. \]

Prove or disprove: \[ \{f:\mathbb H\to \mathbb H\mid f\text { is holomorphic}\} \] is a normal family.