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

Let \(\mu \) be the outer measure on \(\mathbb R\) defined by \[ \mu (A)=\inf \left \{ \sum _{i=1}^{\infty }(b_i-a_i)\,\middle |\, A\subseteq \bigcup _{i=1}^{\infty }(a_i,b_i],\ a_i\le b_i \right \} \] for every \(A\subset \mathbb R\). Prove that \(\mu ((a,b])=b-a\). You may use that the length of a segment covered by finitely many segments is less than the sum of lengths of segments in the covering.

Let \((X,\mathcal M,\mu )\) be a measure space. Assume there exists \(\delta >0\) such that for every \(E\in \mathcal M\), either \(\mu (E)=0\) or \(\mu (E)>\delta \). Let \(f\in L^1(X)\). Prove that \(f\) is essentially bounded.

Let \(X\) be a normed space, and let \(X^*\) be the space of bounded linear functionals on \(X\). For every \(x\in X\), let \(x^*\) be the functional on \(X^*\) defined by \(x^*(y)=y(x)\) for all \(y\in X^*\). Prove that \(x\mapsto x^*\) is an isometry from \(X\) into \(X^{**}\).

Let \(V\) be a normed space and \(U\) a closed subspace. Define the norm on \(V/U\) and prove that \(V/U\) is complete if \(V\) is.

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 \(v\in V\).

Using the methods of complex analysis, compute \[ \int _0^\infty \frac {x^2}{x^4+1}\,dx \] and simplify your answer.

Show that if \(f:\mathbb D\to \mathbb D\) is holomorphic then there exist \(n_1,n_2,\ldots \) so that \(f^{n_i}(z)\) converges pointwise for all \(z\in \mathbb D\). Note that \(f^{n_i}\) denotes \[ \underbrace {f\circ \cdots \circ f}_{n_i\ \text {times}}. \]

Is there an entire function so that \[ f\left (\frac {1}{p}\right )=\frac {1}{1+p} \] for all primes \(p\)?

Show that if \(f:\mathbb D\to \mathbb C\) is holomorphic and \(f'(0)\ne 0\) then there is a neighborhood of \(0\) where \(f\) is injective. Do not use the inverse function theorem.

State and prove Schwarz’s lemma.