DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN ANALYSIS January 2009

Decide whether the following statement is true or false. If true, prove it, if false, provide a counterexample.

Let \(\{f_n\}\) be a sequence of measurable functions (with respect to Lebesgue measure) with \(0\le f_n\le 1\) such that \(\int _0^1 f_n(x)\,dx\to 0\). Then \(f_n\to 0\) almost everywhere.

Let \((X,\mu )\) be a measure space with \(\mu (X)=1\), and let \(f\in L^\infty (X)\). Prove that \(\lVert f\rVert _p\to \lVert f\rVert _\infty \) where \(\lVert f\rVert _p\) denotes the \(L^p\) norm of \(f\).

Let \(X\) and \(Y\) be Banach spaces, and suppose that \(T:X\to Y\) is a compact surjective linear map. Show that \(Y\) is finite dimensional. (Recall that a linear map is compact if the image of any bounded set has compact closure.)

Let \(H\) be a Hilbert space and suppose that \(\{v_n\}\) is a sequence in \(H\) with the following property. For each \(u\in H\) \[ \sup _n |\langle v_n,u\rangle |<\infty \] Prove that \(\sup _n\lVert v_n\rVert <\infty \).

Let \(f\in L^1(\mathbb R)\) such that \[ \int _{-\infty }^{\infty }|x\hat f(x)|\,dx<\infty \] Prove that there is a differentiable function \(g\) on \(\mathbb R\) such that \(f=g\) almost everywhere.

Let \(f\) be a holomorphic function on \(\mathbb C\setminus \{0\}\) such that \(f(z)\sin z\to 1\) as \(z\to 0\). Calculate \[ \int _\gamma f(z)\,dz \] where \(\gamma \) is the unit circle traversed counter-clockwise.

Let \(f\) be a meromorphic function on \(\mathbb C\) such that \(|f(z)|<B\) if \(|z|>R\). Show that \(f\) is a rational function (a quotient of polynomials).

Let \(f:\Omega \to \mathbb C\) be a complex-valued function on an open domain \(\Omega \subset \mathbb C\). Assume that for all closed contours \(\gamma \) that consists of line segments parallel to the real or imaginary axes \[ \int _\gamma f(z)\,dz=0. \] Show that there exists a holomorphic function \(F:\Omega \to \mathbb C\) such that \(F'(z)=f(z)\).

Let \(Q\) be a rectangle in \(\mathbb C\) and \(f:\mathbb C\to \mathbb C\) a holomorphic map. Assume that there exists a \(z\in Q\) such that \(f(z)=z\) and \(f'(z)=1\). Show that \(f\) is the identity map.

Suppose \(a\) is a real number greater than \(1\). Compute \[ \int _0^{2\pi }\frac {d\theta }{a+\sin \theta } \] using residues. Hint: \(\sin \theta =\frac {z-z^{-1}}{2i}\) where \(z=e^{i\theta }\).