Qualifying Examination, August 14, 2002 Do at least seven problems, making sure to do at least three from each set. Real Analysis

Let \(\alpha (x,y)\) be a continuous function on \([0,1]\times [0,1]\). Show that \[ A(f)(x)=\int _0^1 \alpha (x,y)f(y)\,dy \] defines a bounded operator from \(L^2([0,1])\) to \(C([0,1])\), considered as a Banach space with respect to \(\|f\|=\sup _x |f(x)|\).

Let \(P\) be a self-adjoint operator on a Hilbert space \(H\) such that \(P^2=P\). Show that \(P\) is a projection on a closed subspace. (Warning: we do not assume, apriori, that \(P\) is bounded).

Consider \[ HC=\{(x_n)\in \mathbb C^{\mathbb Z}\mid |x_n|\le \frac {1}{n}\text { for all }n\in \mathbb Z\}. \] Note that \(HC\) is simply a product of (infinitely many) closed discs. As such, it is compact with respect to the product topology. The product topology on \(HC\) is defined so that sets \[ U_x(m,\delta )=\{(y_n)\in HC\mid |x_n-y_n|<\delta \text { for all }1\le n\le m\} \] provide a fundamental system of neighborhoods of \(x\). Next, note that \(HC\) can be considered a subset of the Hilbert space \(\ell ^2(\mathbb Z)\). Let \(B(x,\epsilon )\) be, as usual, the open ball of radius \(\epsilon \) in \(\ell ^2(\mathbb Z)\). Show that for every \(\epsilon >0\) there exists a positive integer \(m\) and \(\delta >0\) such that \[ U_x(m,\delta )\subseteq B(x,\epsilon )\cap HC \] and, for every positive integer \(m\) and \(\delta >0\) there exists \(\epsilon >0\) such that \[ U_x(m,\delta )\supseteq B(x,\epsilon )\cap HC \] In other words, show that the restricted topology on \(HC\) coincides with the product topology.

Let \(X\) be a measure space with \(\mu (X)=1\). Let \(f\) be a measurable function in \(L^\infty (X)\). Note that \(\|f\|_p\le \|f\|_\infty \) for all \(p\ge 1\). In particular, \(f\) can be considered an element of \(L^p(X)\) for all \(p\ge 1\). Show that \[ \lim _{p\to \infty }\|f\|_p=\|f\|_\infty . \]

Show that the parallelogram law fails for \(L^1([0,1])\) so it is not a Hilbert space.

Hint: Use \(f=\chi _{[0,1/2]}\) and \(g=\chi _{[1/2,1]}\).

Evaluate the integral: \[ \int _0^\infty \frac {dx}{1+x^3} \]

Let \(B\) and \(C\) be two discs of the same radius which are tangent at a point in \(\mathbb C\). Let \(A\) be another disc of the same radius whose boundary goes through the point of tangency of \(B\) and \(C\). Find a Riemann mapping of \(A-(B\cup C)\) onto the unit disc.

Let \(f\) be a complex valued function defined on the domain \(D\subset \mathbb C\). Suppose, that as a mapping of \(\mathbb R^2\) to \(\mathbb R^2\), \(f=u+iv\) is twice differentiable, and that the jacobian \[ \begin {pmatrix} \dfrac {\partial u}{\partial x} & \dfrac {\partial u}{\partial y}\\[0.6ex] \dfrac {\partial v}{\partial x} & \dfrac {\partial v}{\partial y} \end {pmatrix} \] is everywhere, as a linear transformation, a composition of a dilation and a rotation. Show that \(f\) is a holomorphic function.

Let \(D\) be a domain in the plane, and \(\{f_n\}\) a sequence of functions holomorphic on \(D\). Suppose the sequence is uniformly bounded; that is, there is a positive number \(M\) such that \[ |f_n(z)|<M \quad \text {for all } n \text { and } z\in D. \] Suppose, in addition, that there is an infinite compact set \(K\) in \(D\) such that the sequence \(f_n|K\) converges uniformly on \(K\). Show that the sequence \(f_n\) converges uniformly on all compact subsets of \(D\).

Suppose that \(f\) is meromorphic on the plane, is never zero on the plane, and that \[ \lim _{z\to \infty }|f(z)|=0. \] Show that \(f\) is the inverse of a polynomial.