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\centerline{\Large Math 675 Potential Midterm Exam Problems}
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The exam will consist of 10 problems, separated into definitions, examples, and proofs. Here are some potential questions. 

The list of questions may be modified before the actual exam is compiled.  You should think about what additional problems might be added.

\subsection*{Definitions}
\begin{enumerate}[1.]
\item Metric
\item Closure
\item Vector space
\item Contact point
\item Cauchy
\item Complete
\item Sequentially compact
\item Compact
\item Separable
\item Open
\item Closed
\item Continuous
\item Homeomorphism
\item Isometry
\item Completion
\item Contraction
\item Linearly independent
\item Basis
\item Span
\item Linear functional
\item Null space
\item Hyperplane
\item Codimension
\item Convex
\item Convex body
\item Convex functional
\item Quotient space
\item Minkowski functional
\item Subspace
\item $d_p$
\item $C_p[a,b]$
\item $\ell_p$
\end{enumerate}

\subsection*{Proofs}
\begin{enumerate}[1.]
\item For $a<b$ in $\R$, the closure of $(a,b)$ is $[a,b]$.
\item In $\R^n$, the metric $d_\infty$ is the limit of the functions $d_p$ as $p\rightarrow \infty$.
\item The spaces $C_\infty[a,b]$ and $C_\infty[c,d]$ are isometric for any $a<b$ and $c<d$.
\item A point $x\in X$ is a limit point of $A$ if and only if for every $\epsilon>0$ the ball $B_\epsilon(x)$ contains a point of $A$ not equal to $x$.
\item The closure of $A$ equals the union of the set of isolated points $A$ with the set of limit points of $A$.
\item What is the closure of $\Q\subset \R$? Prove that you are correct.
\item The following are equivalent for a function $f: X\rightarrow Y$ between two metric spaces:
\begin{enumerate}[(i)]
\item $f$ is continuous,
\item $f^{-1}(A)$ is an open set for every open set $A\subset Y$,
\item $f^{-1}(B)$ is a closed set for every closed set $B\subset Y$.
\end{enumerate}
\item  The set of invertible 2-by-2 matrices is open. (Interpret the space of all 2-by-2 matrices as $\R^4$ with the usual topology.) You may use your favorite definition of continuity.
\item The set of determinant-one 2-by-2 matrices is closed. You may use your favorite definition of continuity.
\item Let $A$ be a linearly independent set in a metric vector space $V$. Assuming that $V$ is finite-dimensional, prove that the set $\R A=\{\lambda a : \lambda \in \R, a\in A\}$ is closed.
\item Suppose two metric spaces $X$ and $Y$ are bi-Lipschitz equivalent. Prove that that $X$ is complete if and only if $Y$ is complete.
\item The image of a compact set under a continuous function is compact.
\item Let $f: X\rightarrow X$ be a contraction, and $Y\supset X$ a completion of $X$. Prove that $f$ extends to a contraction of $Y$. Use this to conclude that there is no homeomorphism of $(0,1)$ that is also a contraction.
\item  The contraction theorem requires a mapping that is $L$-Lipschitz for $L<1$. Prove that the result is sharp. That is, allowing $L=1$ in the theorem would make it false.
\item A closed subset of a compact space is compact. 
\item Suppose $f, f_1, f_2$ are linear functionals such that $f(x)=0$ if $f_1(x)=0$ and $f_2(x)=0$. Prove that there are constants $a_1$ and $a_2$ such that $f=a_1 f_1+a_2f_2$.
\item If $X$ is compact and $f: X\rightarrow \R$ is continuous, then $f$ attains a maximum value.
\item If $A, B\subset X$ are compact subsets of a metric space $X$, then $A\cup B$ is compact.
\end{enumerate}

\subsection*{Examples}
\begin{enumerate}[1.]
\item A bounded set that is not compact.
\item A metric vector space that is not separable.
\item A linearly independent set $A$ in a metric vector space $V$ such that the set $\R A=\{\lambda a : \lambda \in \R, a\in A\}$ is not closed.
\item A union of closed sets that is not closed.
\item An intersection of open sets that is not open.
\item Homeomorphic metric spaces $X$ and $Y$ such that $X$ is complete but $Y$ isn't.
\item A compact set $C$ and a continuous function $f$ such that $f^{-1}(C)$ is not compact.
\item A contraction that does not have a fixed point.
\item Let $V$ be a non-trivial vector space. Give an example of a non-trivial functional on $V^*$.
\item A convex set that is not a convex body.
\item Two sets that cannot be separated by a linear functional.
\item A Cauchy sequence in $C_2[a,b]$ that does not converge to a function in $C_2[a,b]$.
\end{enumerate}





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