Prove that a closed subspace of a complete metric space is complete with respect to the induced metric. Conclude that \((c_0, \norm{\cdot}_\infty)\) is a Banach space. (See Exercise 3.3.6.)
2.
Show that the sequence \((f_n)\) in \(C[0,1]\) given by the graph
is Cauchy but not convergent in \(C[0,1]\text{.}\)
3.
Show that the normed space \(C(X), \norm{\cdot}_\infty\) of Checkpoint 3.1.4 is a Banach space.
4.
Show that that the space \(W[a,b]\) of continously differentiable functions taking complex values on \([a,b]\) with inner product (2.3.2) is not a Hilbert space (use the indefinite integrals of the functions in Exercise 4.4.2).
5.
Show that \(RH^2\) is not a Hilbert space.
6.
For which real \(\alpha\) does the function
\begin{equation*}
f_\alpha(t) = t^\alpha e^{-t}, \,\,\, t > 0
\end{equation*}
belong to \(L^2(0,\infty)\text{?}\) When defined, what is \(\norm{f_\alpha}\text{?}\)
7.
Prove that the open and closed (unit) balls in a normed space are convex (see Exercise 3.3.8).
8.
Prove that the closure of a convex set in a normed space is convex.
Show that \(E\) does not have the closest point property by finding infinitely many points in the closed convex set
\begin{equation*}
A = \{(x_1, x_2): x_1 \geq 1\}
\end{equation*}
which are at minimal distance from the origin.
10.
Let \(A\) be a non-empty closed convex set in a Hilbert space. Show that \(A\) contains a unique vector \(a\) of smallest norm that that \(\RE \ip{a}{a-x} \leq 0\) for all \(x \in A\text{.}\)
is Cauchy but not convergent in \(C[0,1]\text{.}\)
