Department of Mathematics University of Utah Ph.D. Preliminary Examination in Analysis May 2018

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 \(V\) be a normed space and \(U\) a closed subspace. For every \(x\in V\), let \[ \lVert x+U\rVert =\inf \{\lVert x+y\rVert :\ y\in U\}. \]

(a) Prove that \(\lVert x+U\rVert \) is a norm on \(V/U\).

(b) If \(V\) is complete, so is \(V/U\).

Let \(T:V\to U\) be a bounded linear map between two normed spaces. Let \(T^*:U^*\to V^*\) be defined by \(T^*(f)=f\circ T\) for all \(f\in U^*\) (the adjoint map). Prove that \(\lVert T^*\rVert =\lVert T\rVert \).

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. Let \(p\ge 1\). For \(n=1,2,\ldots \), let \[ f_n=n^{1/p}\cdot \chi _{[0,1/n]}. \] Prove that the sequence \(f_n\) converges weakly to \(0\) in \(L^p([0,1])\) if and only if \(p>1\). Use that \[ L^p([0,1])^*\cong L^q([0,1]). \]

Let \(a_j\in \mathbb C\) satisfy \(|a_j|\le 1\) for all \(j\ge 2\). Show that \(f:B(0,\frac {1}{100})\to \mathbb C\) defined by \[ f(z)=z+\sum _{j=2}^{\infty }a_jz^j \] is injective.

Show that \(0\) is in the range of \(e^z-z\).

Let \(f:\mathbb H\to \mathbb D\) be a bijective holomorphic map. Are we guaranteed that \[ -f(\bar z)=-\overline {f(z)}? \] In words, is negative \(f\) at the conjugate input equal to the negative complex conjugate of the original output?

Use the methods of complex analysis to compute \[ \int _{-\infty }^{\infty }\frac {2}{(x^2+4)(x^2+1)}\,dx. \]

If \(f,g\) are entire functions and \(|f(z)|\le |g(z)|\) for all \(z\), then there exists \(c\in \mathbb C\) so that \[ f(z)=c\,g(z). \]