Department of Mathematics University of Utah Ph.D. Preliminary Examination in Real/Complex Analysis August 2000 INSTRUCTIONS

Let \(f\) and \(g\) be two real valued measurable functions on a measurable space \(X\). Show that \[ \{x\in X:\sin f(x)\ge \cos g(x)\} \] is a measurable subset of \(X\).

Let \(\mathbb C^n=\{z:z=(z_1,\ldots ,z_n)\}\) be the \(n\)-dimensional complex space with norm \(\|z\|=\sum _{k=1}^n |z_k|\). Prove that there is no scalar product on \(\mathbb C^n\) with the property that for each \(z\in \mathbb C^n\), \(\|z\|^2=(z,z)\).

Let \(X=[0,1]\), and \(C(X)\) the space of complex valued continuous functions on \(X\), equipped with the sup norm: \(\|f\|_\infty =\sup _{x\in X}|f(x)|\). Let \(f_n\) be a sequence in \(C(X)\). Prove that the following statements are equivalent:

(a) For each bounded linear functional \(\lambda \) on \(C(X)\), \(\lambda (f_n)\to 0\);

(b) \(f_n(x)\to 0\) for each \(x\in X\), and \(\sup _n\|f_n\|_\infty \) is finite.

Let \(X\) be a compact Hausdorff space. Let \(C(X)\) be the space of complex valued continuous functions on \(X\). We assume that \(C(X)\) is equipped with a norm \(\|\cdot \|\), which turns it into a Banach space. Assume moreover, that, for each \(x\in X\) the linear functional \(\lambda _x\), defined by: \(\lambda _x(f)=f(x)\) is bounded on this Banach space. Show that there exist positive constants \(A\) and \(B\), such that for every \(f\in C(X)\) \[ A\|f\|_\infty \le \|f\|\le B\|f\|_\infty . \] Here \(\|f\|_\infty \) denotes, as usual, the sup norm.

Let \(f\) be a continuous positive valued function on \(\mathbb R\), and let \(g\) be the characteristic function of a finite nonempty open interval in \(\mathbb R\). Show that the function \(h=fg\) is in \(L^1(\mathbb R)\), but its Fourier transform \(\hat h\) is not.

Let \(f\) be an automorphism of the unit disc \(|z|<1\). Assume that \(f\) fixes two points. Show that \(f(z)=z\) for all \(z\) in the unit disc.

Determine the number of zeroes of the polynomial \[ z^5+z^3/3+z^2/4+z/3 \] in the region \(1/2<|z|<1\).

Let \(\gamma \) be a closed path in \(\mathbb C\setminus [0,1]\). Show that \[ \int _\gamma \frac {1}{z(1-z)}\,dz=0. \]

Let \(f\) be a holomorphic function on the unit disc. For every \(0\le r<1\) define \[ M(r)=\max _{|z|=r}|f(z)|. \] Show that \(M(r)\) is an increasing, continuous function on \([0,1)\).

Determine the poles and their orders of the function \[ \frac {1}{e^z-1}-\frac {1}{z}. \]