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\textbf{Math 366: Honors Analysis II}
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\textbf{Mid-Term Exam}\quad \textsl{March 9, 2001}

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1. (20 pts) Give precise definitions for the following concepts.

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a) uniformly continuous

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b) convolution

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c) inner product

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d) metric space 

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e) limit point of a set in a metric space

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f) open set in a metric space

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g) closed set in a metric space

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h) compact set in a metric space

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i) complete set in a metric space

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j) a continuous function $f:M\to N$ of metric spaces

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2. (20 pts)

a) State the Weierstrass Approximation Theorem.

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b) State the Arzela-Ascoli Theorem.

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You need complete only 5 of the next 6 proofs.

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3. (15 pts) Prove that if $\{f_n\}\subset C[a,b]$ and $f_n\to f$
uniformly, then $f\in C[a,b]$.

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4. (15 pts) Prove that if $f$ is analytic at~$0$ with radius of
convergence~$R$, then $f$ is analytic at any point $a$ with $|a|<R$.

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5. (15 pts) Prove the Cauchy-Schwartz Inequality.

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6. (15 pts) Let $M$ be the metric space $C[a,b]$ with the sup-norm. Prove
that if $A \subset M$ is compact, then $A$ is equicontinuous.

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7. (15 pts) Prove that a compact subset of a metric space must be bounded.

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8. (15 pts) Prove that if $f:M\to N$ is a continuous function of metric
spaces, then $\forall\,1/m$, $\exists\,1/n$ such that
\[ d_M(x,y)<\frac{1}{n} \Rightarrow d_N(f(x),f(y)) < \frac{1}{m} \]

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