Department of Mathematics University of Utah Ph.D. Preliminary Examination in Analysis August 2019

Let \(f\ge 0\) be an integrable function on a measure space \(X\). Prove that, for every \(\epsilon >0\), there exists a set \(E\subset X\) of finite measure such that \[ \int _E f>\left (\int _X f\right )-\epsilon . \]

Let \(V\) be a normed space and \(U\) a closed subspace. For every \(x\in V\), let \[ \lVert x+U\rVert =\inf \{\lVert x+y\rVert :\ y\in U\}. \] Then \(\lVert x+U\rVert \) is a norm on \(V/U\). (You do not need to prove that.) Assume \(U\ne V\). Prove that the map \(P:V\to V/U\) defined by \(P(x)=x+U\) has operator norm \(1\).

Let \(T:U\to V\) be a linear map between two Banach spaces such that, for any linear functional \(f\) on \(V\), the composite \(f\circ T\) is a continuous functional on \(U\). Prove that \(T\) is continuous. Hint: use the closed graph theorem.

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Let \(X=[0,1]\) with the usual measure. For \(n=1,2,\ldots \), let \(f_n=n^{1/p}\cdot \chi _{[0,1/n]}\). Prove that the sequence \(f_n\) does not converge weakly to \(0\) in \(L^1([0,1])\).

Prove the following special case of Runge’s approximation theorem: If \(f:B(0,4)\setminus (B(0,.1)\cup B(1,.1))\) is holomorphic and \(\epsilon >0\) is given, then there exists a rational function \(q\) so that \(|f(z)-q(z)|<\epsilon \) for all \[ z\in \overline {B(0,2)\setminus (B(0,.2)\cup B(1,.2))}. \]

The following statement is false: If \(\Omega \subset \mathbb C\) is any simply connected set and \(x\in \Omega \), then there exists a unique \(r\) and biholomorphism \(f:\Omega \to B(0,r)\) so that \(f(x)=0\) and \(f'(x)=1\). Fix the statement and provide a proof for the correct statement.

Note: Simply connected implies the set is open and connected.

Prove that if an entire function is not a polynomial then there is a number \(z\) so that the cardinality of \(f^{-1}(z)\) is infinite.

Do not use Picard’s theorems.

Prove that if \(f\) is entire and sends the real line to itself, then \[ f(\bar z)=\overline {f(z)} \] for all \(z\in \mathbb C\). In words, applying \(f\) to the conjugate input gives the complex conjugate of the original output.

Compute \[ \int _0^{2\pi }\frac {d\theta }{4+\cos (\theta )} \] using the methods of complex analysis.