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\centerline{\Large Math 675 Potential Final Exam Problems}
\vspace{.25in}

The exam will consist of 20 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 (set)
\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 $\ell_p$
\item Norm
\item Inner product
\item Bessel's inequality
\item Parseval's identity
\item Fourier coefficient
\item Hilbert space
\item Banach space
\item H\"older's inequality
\item Cauchy's inequality
\item Orthogonal complement
\item Riesz-Fisher Theorem
\item Continuous linear functional
\item Dual space
\item Dual operator
\item Spectrum
\item Adjoint
\item Eigenvalue
\item Eigenvector
\item Completely continuous
\item Compact operator
\item Self-adjoint
\item Weierstrass approximation theorem
\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.
\item Suppose $A,B\subset V$ are separated by a linear functional $f$. If $A$ is a convex body, prove that there are points $a\in A$ and $b\in B$ such that $f(a)\neq f(b)$.
\item Let $\Norm{\cdot}$ be the norm in $\R^2$ for which the unit circle is a regular hexagon with side length 1. Prove that $\Norm{\cdot}$ is not induced by an inner product. (Hint: the parallelogram law states  $2(a^2+b^2)=(a-b)^2+(a+b)^2$).
\item Given a Banach space $V$, let $\{B_n\}$ be a nested sequence of closed spheres in $V$. Prove that $\cap_n B_n$ is nonempty. (Note that the radius is not assumed to go to zero, and the centers are not assumed to be the same.)
\item Let $\{e_i\}$ be an orthonormal basis for a Hilbert space $H$. Take $x_n=e_{2n}$ and $y_n=e_{2n}+\frac{e_{2n+1}}{n+1}$, and set $M=\span(\{x_n\})$, $N=\span(\{y_n\})$. Show that $M+N$ is not closed, even though $M$ and $N$ are both closed.
\item Prove that the functional $F(f)=\int_a^b f(t) \cos(t)dt$ is a continuous functional on $C_\infty[a,b]$. What could we replace $\cos(t)$ with?
\item Prove that both of the following is a subspace of $\ell^2$: the set of all $(x_i)$ such that $x_1=x_2$.
\item Prove that both of the following is a subspace of $\ell^2$: the set of all $(x_i)$ such that  $x_k=0$ for all even $k$.
\item Let $V$ be a normed real vector space and $F: V\rightarrow \R$ a linear functional. Prove that $F$ is continuous if and only if $N(F)$ is closed. (Hint: The forward direction requires no work at all; for the backward direction, assume $F$ is not bounded and show that $N(F)$ is not closed by perturbing a non-zero point.)
\item Prove that  following functional is continuous on $C_\infty[0,1]$ and compute its norm: $f(x)=ax(0)+bx(1)$.
\item Prove that  following functional is continuous on $C_\infty[0,1]$ and compute its norm: $g(x)=\int_0^{1/2}x(t)dt - \int_{1/2}^1 x(t)dt$.
\item Prove that if $p<q<\infty$ and $f$ is a linear functional on $C_p[0,1]$ then it is also continuous on $C_q[0,1]$.
\item Let $V_0$ be a normed real vector space and $V$ its completion. Prove that $V_0^*$ and $V^*$ are  isomorphic  Banach spaces.
\item Recall that a sequence $v_i$ in a normed space $V$ converges weakly to $v$ if $f(v_i)$ converges to $f(v)$ for each linear functional $f\in V^*$. Prove that the standard basis vectors $e_i$ in $\ell_2$ weakly converge to the zero vector.
\item Prove that an infinite-dimensional Banach space has uncountable algebraic dimension.
\item Prove that every element of $C_2[a,b]$ is of the form $\sum_1^\infty a_ip_i(x)$ for $a_i\in \R$ and $p_i$ a polynomial of degree $i$.
\item A linear functional is continuous if and only if it is bounded.
\item Let $V\subset C_\infty[a,b]$ consist of all continuously differentiable functions. Prove that the differentiation operator $D:V\rightarrow C_\infty[a,b]$ is not continuous.
\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 disjoint sets in $\R^2$ 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 $C_2[a,b]$.
\item An explicit linear functional on $\R^2$ whose null space is spanned by the vector $(1,2)$.
\item An explicit convex functional on $\R^2$.
\item A function in $C_1[0,1]$ but not $C_\infty[0,1]$. 
\item A function in $C_1[0,1]$ but not $C_2[0,1]$.
\item A linear function that is continuous on $C_\infty[0,1]$ but not on $C_1[0,1]$.
\item A continuous operator that is not compact.
\item A compact operator that has no eigenvalues.
\item A continuous operator that has no eigenvalues.
\item An operator from a space to itself that is not self-adjoint.
\item An operator $A:\ell_2\rightarrow \ell_2$ whose eigenvalues include $1$ and $2$.
\item A non-trivial operator $A:\ell_2\rightarrow \ell_2$ whose eigenvalues are all smaller than $1/2$.
\item an inner product space $V$ and an orthonormal system $\{e_i\}$ such that $V$ contains no non-zero element orthogonal to every $e_i$, even though $\{e_i\}$ does not span $V$.
\item A nested sequence of closed non-empty sets $C_i$ in $\R$ such that $\cap C_i=\emptyset$.
\item A nested sequence $\{E_n\}$ of nonempty closed bounded convex sets in a Banach space $V$ (of your choice) such that $\cap_n E_n=\emptyset$.
\item A basis for $C_2[0,\pi]$.
\item An element of $\ell_\infty$ that cannot be written as $\sum_i^\infty a_i e_i$, where $a_i\in \R$ and $e_i$ is the sequence $(0,\ldots, 0, 1, 0, \ldots)$ where the 1 is in the $i^{th}$ place.
\item An continuous operator $A$ that is invertible such that $A^{-1}$ is not continuous.
\item Recall that the \emph{support} of a function is the closure of the set of points where it has non-zero values. Suppose $f\in C_2[0,1]$ is supported on $[0,1/2]$. Give an example of a non-trivial element of $f^\perp$.
\item Recall that $V$ is reflexive if the canonical mapping from $V$ to $(V^*)^*$ is an isomorphism. Give an example of a non-reflexive space.
\end{enumerate}





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