REAL AND COMPLEX ANALYSIS QUALIFYING EXAMINATION January, 2010

Let \([0,1]\) be the unit interval equipped with Lebesgue measure and fix \(1\le p<\infty \). Suppose \(f_1,f_2,\ldots \) is a sequence in \(L^p([0,1])\) which is Cauchy with respect to the \(L^p\) norm.

(a) Prove that there is a subsequence \(g_1,g_2,\ldots \) of \(f_1,f_2,\ldots \) such that \(\lim _{i\to \infty }g_i(x)\) exists for almost all \(x\).

(b) Show (by producing an example of a Cauchy sequence \(f_1,f_2,\ldots \) in \(L^p([0,1])\)) that \(\lim _{i\to \infty }f_i(x)\) can fail to exist for almost all \(x\).

Suppose \((X,\mathcal M,\mu )\) is a measure space and that \(L^p(X)\subset L^q(X)\) for some \(1\le p<q<\infty \).

(a) Show that the inclusion \(L^p(X)\to L^q(X)\) is bounded.

(b) Let \(\mathcal M'\) denote the subsets in \(\mathcal M\) of nonzero measure. Use (a) to prove that \[ \inf _{E'\in \mathcal M'}\mu (E')>0. \]

Let \(H\) be a Hilbert space, and let \(V\subsetneq H\) be a nonzero proper closed subspace. Let \(\pi _V\) be the orthogonal projection onto \(V\).

(a) Show that \(\pi _V\) has norm \(1\), is idempotent (\(\pi _V^2=\pi _V\)), and is self-adjoint (\(\pi _V^*=\pi _V\)).

(b) Conversely, if \(P:H\to H\) is a self-adjoint, idempotent, continuous linear transformation of operator norm \(1\), show that \(P=\pi _V\) for some closed subspace \(V\) of \(H\).

(a) Let \(\ell ^1\) and \(\ell ^\infty \) be the Banach spaces of sequences of complex numbers \(\xi =(\xi _1,\xi _2,\ldots )\) such that \[ \xi \in \ell ^1 \quad \text {if and only if}\quad \lVert \xi \rVert _1:=\sum _{i=1}^{\infty }|\xi _i|<\infty \] and \[ \xi \in \ell ^\infty \quad \text {if and only if}\quad \lVert \xi \rVert _\infty :=\sup _i|\xi _i|<\infty . \] Let \((\ell ^\infty )^*\) denote the Banach space of continuous linear functionals, equipped with the operator norm, on \(\ell ^\infty \). Show that the map \(\Lambda :\ell ^1\to (\ell ^\infty )^*\) defined by \[ \Lambda (\xi )(\eta _1,\eta _2,\ldots )=\sum _i \xi _i\eta _i \] is a well-defined norm-preserving injection of \(\ell ^1\) into \((\ell ^\infty )^*\) which is not onto.

Hint: Show that \((\ell ^\infty )^*\) contains nontrivial functionals which vanish on the subspace consisting of those \(\xi \) such that \(\xi _i\to 0\) as \(i\to \infty \).

(b) Let \((X,\mu )\) be a measure space. State precise conditions on \(\mu \), \(p\), and \(q\) which guarantee a norm-preserving isomorphism of \(L^p(X)\) onto \((L^q(X))^*\).

Let \((X,\mathcal M,\mu )\) be a measure space such that \(\mu (X)<\infty \). Let \(f:X\to X\) be measure preserving in the sense that \(\mu (E)=\mu (f(E))\) for all \(E\in \mathcal M\). Prove that, for any \(E\in \mathcal M\), the set of points \(x\) of \(E\) such that the \(n\)th iterate \(f^n(x)\) is not in \(E\) for all \(n>0\) has zero measure.

Hint: Suppose not and obtain a contradiction with the fact that \(\mu (X)<\infty \).

Let \[ f(z)=\frac {6z+2}{(z^2-1)(z+3)}. \]

(a) Use the residue theorem to compute \(\int _\gamma f(z)\,dz\), where \(\gamma \) is the positively oriented circle \(|z|=2\).

(b) Compute the Laurent series for \(f(z)\) which converges in the annulus \(1<|z|<3\). Use this to check your computation of the integral in (a).

Let \(n\) be a positive integer. Use a contour integral to compute \[ \int _0^{2\pi }\cos ^{2n}(\theta )\,d\theta \] in terms of \(n\).

Let \(U\) be an open connected domain in the complex plane, and let \(f\) be analytic on \(U\).

(a) Prove or disprove: if \(f\) is injective on \(U\), then the derivative \(df/dz\) is never zero on \(U\).

(b) Prove or disprove: if the derivative \(df/dz\) is never zero on \(U\), then \(f\) is injective on \(U\).

Find all conformal transformations \(f(z)\) from the upper half disc \[ \{z\mid |z|<2 \text { and } \operatorname {Im}z>0\} \] onto the unit disc \(\{z\mid |z|<1\}\) such that \(f(i)=0\).

Let \(f(z)\) be an analytic function in the unit disc \(\{z\mid |z|<1\}\) such that \(f(z)\) is bounded in modulus by \(M\): \(|f(z)|\le M\). Let \(0<r<1\). Find a constant \(C\) depending on \(M\) and \(r\) so that for all \(z,w\) of modulus less than \(r\), \[ |f(z)-f(w)|\le C|z-w|. \]