Department of Mathematics University of Utah Ph.D. Preliminary Examination in Real/Complex Analysis Fall 1996 INSTRUCTIONS

Let \((X,\mathfrak M,\mu )\) be a measure space, and let \(A_1\supset A_2\supset \cdots \supset A_n\supset \cdots \) be a sequence of sets in \(\mathfrak M\).

Is it always true that \(\mu (\bigcap _{n=1}^{\infty }A_n)=\lim _{n\to \infty }\mu (A_n)\)? If so, give a proof. Otherwise give a counterexample.

Let \(H\) be a Hilbert space and let \(u\) and \(v\) be two distinct elements of \(H\), both of norm \(1\).

Show that \(\frac {u+v}{2}\) has norm strictly less than \(1\). Conclude that the unit sphere \(S=\{w\in H:\lVert w\rVert =1\}\) cannot contain a nontrivial interval (i.e. an interval with distinct endpoints).

Let \(\ell ^\infty \) be the normed space consisting of all complex valued, bounded sequences \[ \mathbf a=(a_1,a_2,\cdots a_n\cdots ) \] with the norm \(\lVert \mathbf a\rVert =\sup \{|a_n|:n\in \mathbb N\}\). Let \(W\subset \ell ^\infty \) be the subspace consisting of all \(\mathbf a\in \ell ^\infty \) such that \(\{n\in \mathbb N:a_n\ne 0\}\) is a finite set.

Show that there exists a nontrivial bounded linear functional on \(\ell ^\infty \) which vanishes on \(W\).

Let \(f\in L^1(\mathbb R)\). Define a function \(\varphi \) on \(\mathbb R\) by the formula \(\varphi (x)=\int _{-\infty }^{x}f(t)\,dt\). Show that \(\varphi \) is a uniformly continuous function on \(\mathbb R\).

Using the definitions of convolution and of Fourier transform show that, for \(f,g\in L^1(\mathbb R)\), \[ \widehat {(f*g)}(t)=\hat f(t)\hat g(t) \] for \(t\in \mathbb R\).

(a) Let \(f\) be an entire function. Assume that there exists a polynomial \(g\) such that for each \(z\in \mathbb C\), \(|f(z)|\le |g(z)|\). Show that \(f\) is also a polynomial.

(b) Can we draw the same conclusion if the function \(g\) is rational? In other words, let \(f\) be an entire function and \(g\) a rational function such that for each \(z\in \mathbb C\), which is not a pole of \(g\), \(|f(z)|\le |g(z)|\). Does it follow that \(f\) is a polynomial?

Let \(f\) be an entire function such that \(f(\mathbb C)\subset \mathbb C-S^1\), where \(S^1\) is the unit circle. Show that \(f\) is a constant function.

Let \(D\) be an open set in \(\mathbb C\). Let \(\mathcal F\) be a collection of holomorphic functions defined on \(D\).

(a) Assume \(\mathcal F\) is uniformly bounded on the boundary \(\partial \Delta \) of each triangle \(\Delta \) contained in \(D\). Show that \(\mathcal F\) forms a normal family.

(b) Is \(\mathcal F\) a normal family if we assume it is uniformly bounded when restricted to any interval contained in \(D\)?

Evaluate \[ \int _{\Gamma }\frac {\cos z}{z^5}\,dz \] where \(\Gamma \) is a positively oriented simple unit circle \(\{z:|z|=1\}\).

Let \(P(z)=kz+z^k\) where \(k\) is a positive integer. Show that \(P\) is one-to-one on \[ \{z:|z|<1\}. \]