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\textbf{Math 366: Honors Analysis II}
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\textbf{Final Exam}\quad \textsl{May 11, 2001}

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1. (30 pts) Give precise definitions for the following concepts.

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a) uniform convergence

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b) convolution

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c) metric space

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d) compact set

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e) complete set

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f) connected set and arc-wise connected set

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g) Borel sets

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h) Lebesgue measure

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i) measurable function

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j) Lebesgue integral

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2. (50 pts) Give precise statements of the following theorems.

a) Weierstrass Approximation Theorem

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b) Arzela-Ascoli Theorem

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c) Contractive Mapping Principle

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d) Implicit Function Theorem

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e) Lebesgue Monotone Convergence Theorem

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3. (10 pts) Prove that $A$ is open if and only if $A^c$ is closed.

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4. (10 pts) Prove that if $M$ is a compact metric space and $f:M\to\R$
is continuous, then $f$ is uniformly continuous.

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5. (10 pts) Prove that if $f$ is analytic on $(a,b)$ then the values
of $f$ on $(a,b)$ are determined by its values on any subinterval
$(c,d)\subset (a,b)$.

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6. (10 pts) Let $M$ be the set of continuous functions $f:[0,1]\to[0,1]$.
Prove that $M$ with the sup-norm is a complete metric space.

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7. (10 pts) Let $M$ be as in problem~6 and consider the transformation
  \[ Tx(t) = \frac{1}{2} + \frac{1}{4}\int_0^t x(s)(1-x(s))\,ds \]
Prove that $T:M\to M$ and is a contractive mapping. Then find the fixed
point of~$T$.

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8. (10 pts) Determine which level curves of $F(x,y) = x^2 + y^2 - x^4
- y^4$ have singular points and where those singular points occur. Then
describe how the level curves $F(x,y) = c$ vary as $c$ varies.

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9. (10 pts) Define the Cantor set. Prove that it is uncountable and
has measure~0.

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10. (10 pts) Prove the following statements about measurable sets with
respect to a measure~$\mu$.

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\begin{enumerate}

\item If $A\subset B$ then $\mu(A) \le \mu(B)$.

\item If $A_1 \subset A_2 \subset A_3 \dots$, then
$\mu\big(\bigcup_{j=1}^\infty A_j\big) = \lim_{j\to\infty} \mu(A_j)$.

\end{enumerate}

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11. (10 pts) Prove the following statements about Lebesgue integrals
of measurable functions on measurable sets.

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\begin{enumerate}

\item If $\mu(A) = 0$ and $f\in\LL_\mu(A)$ then $\int_A f\,d\mu = 0$.

\item If $f,g\in\LL_\mu(A)$ and $f = g$ almost everywhere on $A$ then
$\int_A f\,d\mu = \int_A g\,d\mu$.

\end{enumerate}

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12. (10 pts) Define Hausdorff measure, $\mu^\alpha$, on Borel sets. Prove
that that $\mu^0$ is counting measure and $\mu^1$ is Lebesgue measure.

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