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

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 \((X,\mu )\) be a (not necessarily finite) measure space and \(f\ge 0\) be a \(\mu \) integrable function. Prove that, for every \(\epsilon >0\), there exists a set \(E\subset X\) of finite measure such that \[ \int _E f\,d\mu >\int _X f\,d\mu -\epsilon . \]

Let \(T:U\to V\) be a linear map between two Banach spaces such that, for any linear functional \(f\) on \(V\), the composite \(f\circ T\) is a continuous functional on \(U\). Prove that \(T\) is continuous. Hint: use closed graph theorem.

Let \(H=L^2(\mathbb R/\mathbb Z)\), with the usual Lebesgue measure on \(\mathbb R/\mathbb Z\) of total measure \(1\). It is well known that this space has a Hilbert basis \(e_n=e^{2\pi i n x}\), \(n\in \mathbb Z\). Let \(f\in H\) be given by \(f(x)=x\) on \([0,1)\). Compute \(\lVert f\rVert _2^2\) directly and then using Parseval’s equality.

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Show that all the singularities of the function \(f(z)=1/(e^z+3z)\) have order \(1\). You do not need to find the singularities.

Either prove that there is no entire function \(f\) so that \(f(n)=0\) for all \(n\in \mathbb N\) and \(f(z)\ne 0\) for all \(z\in \mathbb C\setminus \mathbb N\), or give an example of such a function with justification.

Prove or disprove: Let \[ \Omega =\{z:\ |\operatorname {Re}(z)|<1\text { and }|\operatorname {Im}(z)|<1\}. \] If \(f:\mathbb D\to \Omega \) is a biholomorphism so that \(f(z)\) is real for all real \(z\in \mathbb D\) and \(f(0)=0\), then \[ f(iz)=i f(z). \]

How many zeros (with multiplicity) does \[ z^6+5z^4+1 \] have in \(B(0,2)\)?

State and prove Morera’s Theorem.