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

Let \(\mu \) be the outer measure on \(\mathbb R\) defined by \[ \mu (A)=\inf \left \{ \sum _{i=1}^{\infty }(b_i-a_i)\ \middle |\ A\subset \bigcup _{i=1}^{\infty }(a_i,b_i],\ a_i\le b_i \right \} \] 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 \(f:[0,\infty )\to \mathbb R\) be defined by \(f(x)=e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral.

Let \(T:V\to W\) be a bounded linear map, where \(V\) and \(W\) are normed spaces. Let \(U\) be the kernel of \(T\). Prove that \(\lVert S\rVert =\lVert T\rVert \) where \(S:V/U\to W\) is such that \[ S(x+U)=T(x) \] for all \(x\in V\).

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 \(H\) 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.

Notation: \(D=\{z\in \mathbb C:|z|<1\}\).

Calculate \[ \int _\gamma (1+z^2)\sin (1/z)\,dz \] where \(\gamma \) is the unit circle traversed in the counter-clockwise direction.

Let \(f\) be a nonconstant entire function. Assume that the set \[ \{z\in \mathbb C:|f(z)|<1\} \] is unbounded. Show that \(f\) has an essential singularity at \(\infty \).

Let \(f:D\to D\) be holomorphic. Show that \[ |f(z)-f(0)|\le |z(1-\overline {f(0)}f(z))| \] for all \(z\in D\). In words, the coefficient multiplying \(f(z)\) in the parentheses is the complex conjugate of \(f\) at zero.

Show that the equation \[ e^z=2z+1 \] has exactly one solution in \(D\).

Hint: Note that \[ e^z-1=\int _0^1 e^{tz}z\,dt. \] Conclude that \(|e^z-1|\le e-1\).

Find a biholomorphism \(\varphi \) of \(D\setminus \mathbb R^-\) onto \(D\). Here \(\mathbb R^-=\{x\in \mathbb R:x\le 0\}\).

(It will suffice to give explicitly finitely many maps whose composition is equal to \(\varphi \).)