DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN ANALYSIS August 2015

Prove the Baire category theorem. That is, show that if \((X,d)\) is a complete metric space and \(U_1,U_2,\ldots \subset X\) are open and dense, then \(\cap _{i=1}^{\infty }U_i\ne \emptyset \).

(a) If \(f_i\to g\) are in \(L^1(\mathbb R,\lambda )\) and \(\phi \in C(\mathbb R)\), does \[ \lim _{i\to \infty }\int f_i\phi \,d\lambda =\int g\phi \,d\lambda ? \]

(b) Show that if \(f_i\to g\) in \(L^1(\lambda ,[0,1])\), then there exists a subsequence \(n_1,\ldots \) so that \(f_{n_i}\to g\) almost everywhere.

If \(v_i\in \mathcal H\) converges weakly to \(v_\infty \) and \(\lVert v_\infty \rVert =\lim _{i\to \infty }\lVert v_i\rVert \), then does \(v_i\) converge in norm to \(v_\infty \)?

Let \(f\) be in \(L^1(\lambda ,\mathbb R)\). Show that for any \(r>0\) the function \[ h(s)=\int _{s-r}^{s+r} f\,d\lambda \] is continuous.

(a) Let \(A\subset [0,1]\) and \(A\in \mathcal B\). Show that for all \(\epsilon >0\) there exists a finite number of intervals \(I_1,\ldots ,I_n\) so that \[ \lambda \left (A\Delta \left (\cup _{j=1}^n I_j\right )\right )<\epsilon . \] Note: \(\Delta \) denotes symmetric difference. So \[ A\Delta B=\{x\in A\cup B:x\notin A\cap B\}. \]

(b) Show that we cannot assume that \(A\subset \cup _{j=1}^n I_j\).

Complex Analysis

Let \(\mathbb D\) denote the open unit disk in \(\mathbb C\). A region is an open connected subset of \(\mathbb C\).

Prove or give a counterexample: If \(\Omega \) is a region, \(f:\Omega \to \mathbb C\) admits unrestricted analytic continuation on a region \(D\supset \Omega \), then there exists \(F:D\to \mathbb C\), holomorphic and with \(F|_\Omega =f\).

Recall: If \(f:\Omega \to \mathbb C\) is an analytic function, we say \(f\) admits unrestricted analytic continuation on \(D\) if for any \(\gamma :[0,1]\to D\), a piecewise smooth path, there exists an analytic continuation of \((f,\Omega )\) along the curve \(\gamma \).

Classify entire biholomorphisms. That is, what are all \(f:\mathbb C\to \mathbb C\) holomorphic with a holomorphic inverse \(g:\mathbb C\to \mathbb C\)?

Does there exist \(f:\mathbb D\to \mathbb C\) holomorphic so that \[ f\left (\frac {1}{n}\right )f\left (\frac {1}{n+1}\right )=\frac {1}{n} \] for all \(n\in \{2,3,\ldots \}\)?

Let \(\mathcal F\) be a family of holomorphic functions on \(\mathbb D\) so that there exists \(C\) with \(|f(z)|<C\) for all \(z\in \mathbb D\).

(a) If \(D\subset \mathbb D\) satisfies that the closure \(\overline D\) is contained in \(\mathbb D\), then \(\mathcal F\) is uniformly Lipschitz on \(D\). That is, there exists \(u\) so that for any \(f\in \mathcal F\) and \(w,z\in D\) we have \[ |f(z)-f(w)|<u|z-w|. \]

(b) Show that \(\mathcal F\) is not necessarily a uniformly Lipschitz family on \(\mathbb D\).

What is \[ \int _{-\infty }^{\infty }\frac {(\cos x)^2}{x^2+5}\,dx? \]