PhD. Analysis Exam
Do 10 of the following 15 problems.
1. Suppose {fn}is a sequence of measurable real valued functions. Define A≡ {x:{fn(x)}converges.}
Is Ameasurable? Explain why or give a counter example.
2. A sequence {xn}∞
n=1 of points in a Banach space, Xis weakly bounded if for every x∗∈X0,the set of
complex numbers, {x∗(xn)}∞
n=1 is bounded. Show that a weakly bounded sequence is in fact bounded.
3. State the open mapping theorem and using this theorem, give a proof of the closed graph theorem.
4. Suppose (Ω,F,µ) is a measure space and let f: Ω →Rbe a measurable function. Suppose g:R→R
is Borel measurable. Does it follow that g◦fis measurable? Give either a proof or a counter example.
5. Give an example in which the Vitali convergence theorem applies but the Dominated convergence
theorem does not apply.
6. The maximal function of f∈L1(Rn) is given by
Mf (x)≡sup (1
mn(B(0,r)) ZB(x,r)
|f(y)|dmn(y) : r > 0).
Using some version of the Vitali covering theorem or other method, establish the weak (1,1) estimate,
mn({x:|Mf (x)|> δ})<Cn
δ||f||L1(Rn)
where Cis some constant which is independent of n. Here mnis the outer measure determined by n
dimensional Lebesgue measure.
7. Let {fn}∞
n=1 be a set of functions which are bounded in L5(Ω) where (Ω,F, µ) is a finite measure
space. Suppose also that
lim
n→∞ fn(x) = f(x).
Can you conclude that
lim
n→∞ Zfn(x)dµ =Zf(x)dµ?
Explain why or why not.
8. Let f:R→Rbe everywhere differentiable. Give an example which shows that f0does not need to be
continuous. Show however that f0must be Borel measurable.
9. Using the Cauchy integral formula, give a short proof of the fundamental theorem of algebra which
states that every non constant polynomial has a zero in the complex plane.
10. An entire function, f(z+a) = f(z) and f(z+ib) = f(z) for a, b two positive real numbers. Suppose
also that f(a)=1.Find a formula for f(z) valid for all z∈C. Explain why your formula is correct.
No guessing.
11. Suppose fis an entire function which satisfies |f(z)| ≤ C1 + |z|1/2and f(0) = 0.Find a formula
for f(z) which is valid for all zand justify your answer.
12. Suppose {fn}is a sequence of functions which are analytic on Ω,a bounded region such that each fnis
also continuous on Ω. Suppose that {fn}converges uniformly on ∂Ω.Show that then {fn}converges
uniformly on Ω and that the function to which the sequence converges is analytic on Ω and continuous
on Ω.
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