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\textbf{Math 366: Honors Analysis II}
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\textbf{Quiz 4}\quad \textsl{March 30, 2001}

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In the following problems, $M$ and $N$ are metric spaces.
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1. Define:

a) $A\subset M$ is connected.

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b) $A\subset M$ is arc-wise connected.

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c) $f:M \to N$ is a contractive mapping.

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2.  State:

a) Contractive Mapping Principle

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b) Minkowski's Inequality

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c) Existence and Uniqueness Theorem for Systems of Differential Equations

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Do two of the following:

3. Prove that if $f:M\to N$ is continuous and $A\subset M$ is compact,
then $f(A)\subset N$ is compact.

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4. Show that any $m$-th order system of differential equations is
equivalent to a first order system.

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5. Let $M$ be the complete metric space of continuous functions
$x:[0,1/2]\to[0,1]$ with the sup-norm. For $x\in M$ define
  \[ Tx(t) = \int_0^t 1 + x(s)^2\,ds,\quad t\in[0,1/2] \]
Show that $T:M\to M$ and that $T$ is a contractive mapping. (Bonus: find
the fixed point of $T$.)

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