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\centerline{\Large Math 675 Homework 10}
\centerline{Due 11/14/2018}
\vspace{.25in}

\begin{enumerate}[1.]
\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 Give an example of 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 Prove that both of the following are subspaces of $\ell^2$:
\begin{enumerate}
\item The set of all $(x_i)$ such that $x_1=x_2$. 
\item The set of all $(x_i)$ such that $x_k=0$ for all even $k$.
\end{enumerate}
\item (Optional) Prove that every nested sequence of nonempty closed bounded convex sets in a Hilbert space (of any dimension) has a nonempty intersection. (Do not use anything beyond Chapter 16.)
\end{enumerate}





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