1.
Determine if the following sets \(S\) are open in the subspace \(A\) of the topological space \((\R, d_E)\text{.}\)
(a)
\(S = [1,2)\) in \(A = [1,3]\)
(b)
\(S = \{1, 2\}\) in \(A = \Q\)
(c)
\(S = \{1,2\}\) in \(A = \Z\)
Exercises Exercises
2.
Let \(O\) be an open set in a metric space \((X,d)\text{.}\) Show that a subset \(U\) of \(O\) is open in \((O, d|_O)\) if and only if \(U\) is open in \((X,d)\text{.}\)
3.
Let \((X,d_X)\) and \((Y, d_Y)\) be metric spaces, and let \(f : X \to Y\) be a continuous function. If \(A\) is a subspace of \(X\text{,}\) must the restriction \(f|_A\) of \(f\) to \(A\) mapping \(A\) to \(Y\) be continuous? Give a proof that the restriction is continuous, or an example to show that the restriction need not be continuous.
4.
Prove Theorem 11.3. That is, let \((X,d)\) be a metric space and \(A\) a subset of \(X\text{.}\) Prove that a subset \(C_A\) of \(A\) is closed in \(A\) if and only if there is a closed set \(C_X\) in \(X\) so that \(C_A = C_X \cap A\text{.}\)
5.
Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces. Prove or disprove: the function \(d: X \times Y \to \R\) defined by
\begin{equation*}
d((x_1,y_1), (x_2,y_2)) = d_X(x_1,x_2) + d_Y(y_1,y_2)
\end{equation*}
is a metric on \(X \times Y\text{.}\)
6.
Let \((X_i, d_i)\) be metric spaces for \(i\) from \(1\) to some positive integer \(n\text{.}\) Let \(d: \prod_{i=1}^n X_i \to \R\) be defined
\begin{equation*}
d(x,y) = \sqrt{\sum_{i=1}^n d_i(x_i,y_i)^2}
\end{equation*}
when \(x = (x_1, x_2, \ldots,
x_n)\) and \(y = (y_1, y_2, \ldots, y_n)\) are in \(X\text{.}\) Show that \(d\) is a metric on \(\prod_{i=1}^n X_i\text{.}\)
7.
Let \((x_n)\) be a non-decreasing sequence of real numbers that is bounded above. That is, \(x_n \leq x_{n+1}\) for every \(n\) and there is a positive real number \(K\) such that \(x_n \leq K\) for every \(n\text{.}\) Show that the sequence \((x_n)\) converges.
8.
It is possible to consider infinite products as metric spaces. One important example is a Hilbert space \(H\text{,}\) which consists of all infinite sequences \((x_n)\) where \(x_n \in \R\) for every \(n\) and \(\sum_{k = 1}^{\infty} x_k^2\) is finite. Hilbert space has important applications in physics, particularly in quantum mechanics.
(a)
Give two distinct elements in \(H\) and one infinite sequence that is not in \(H\text{.}\) Explain your examples.
(b)
We define the norm of an element \(x = (x_n)\) in \(H\) as
\begin{equation*}
\lVert x\rVert = \sqrt{\sum_{k=1}^{\infty} x_k^2}\text{.}
\end{equation*}
From this norm we can define a distance between elements \(x = (x_n)\) and \(y = (y_n)\) in \(H\) as follows:
\begin{equation*}
d(x,y) = \lVert x-y \rVert\text{,}
\end{equation*}
where \(x-y = (x_n-y_n)\text{.}\) Another way to write \(d\) is
\begin{equation*}
d(x,y) = \sqrt{\sum_{k=1}^{\infty} (x_k-y_k)^2}\text{.}
\end{equation*}
One potential problem with this function \(d\) is that we need to know that if \(x\) and \(y\) are in \(H\text{,}\) then \(x-y \in H\text{.}\) That is, show that if \(\sum_{k=1}^{\infty} x_k^2\) and \(\sum_{k=1}^{\infty} y_k^2\) are finite, then \(\sum_{k=1}^{\infty} (x_k-y_k)^2\) is also finite.
Hint.
Consider a finite sum and use Exercise 7.
(c)
Show that \(d\) is a metric on \(H\text{.}\)
(d)
Let \(E^m = \{(x_n) \in H \mid x_k = 0 \text{ for } k > m\}\text{.}\) Let \(f: E^m \to \R^m\) be defined by \(f((x_n)_{n=1}^{\infty}) = (x_n)_{n=1}^m\text{.}\) Show that \(f\) is a bijection such that \(d((x_n), (y_n)) = d_E(f((x_n)),
f((y_n)))\) for any elements \((x_n)\text{,}\) \((y_n)\) in \(H\text{.}\) So \(E^m\) is essentially the same as \(\R^m\) and so we can consider the space \(\R^m\) as embedded in \(H\) as a subspace of \(H\) for every \(m \in \Z^+\text{.}\)
9.
For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.
(a)
If \(d\) is the discrete metric on a metric space \(X\text{,}\) then for any subspace \(A\) of \(X\text{,}\) the restriction of \(d\) to \(A\) is the discrete metric.
(b)
If \(d\) is a metric on a space \(X\) that is not the discrete metric, and if \(A\) is a subset of \(X\text{,}\) then \(d|_A\) cannot be the discrete metric.
(c)
Let \(A\) be a subspace of a metric space \((X,d)\text{.}\) If a sequence \((a_n)\) is in \(A\) and \(\lim a_n = a\) for some \(a \in A\text{,}\) then \(\lim a_n = a\) in \(X\text{.}\)
(d)
Let \(A\) be a subspace of a metric space \((X,d)\text{.}\) If a sequence \((a_n)\) is in \(A\) and \(\lim a_n = a\) for some \(a \in X\text{,}\) then \(\lim a_n = a\) in \(A\text{.}\)
(e)
If \((X,d_X)\) and \((Y, d_Y)\) are metric spaces, then the function \(d: X \times Y \to \R\) defined by
\begin{equation*}
d((x_1,y_1), (x_2,y_2)) = \max\{d_X(x_1,x_2), d_Y(y_1,y_2)\}
\end{equation*}
is a metric on \(X \times Y\text{.}\)
(f)
If \((X,d_X)\) and \((Y, d_Y)\) are metric spaces, then the function \(d: X \times Y \to \R\) defined by
\begin{equation*}
d((x_1,y_1), (x_2,y_2)) = d_X(x_1,x_2)d_Y(y_1,y_2)
\end{equation*}
is a metric on \(X \times Y\text{.}\)
