Show that the \(\ell_1\)-norm and the \(\ell_{\infty}\)-norm do define norms on both \(\mathbb R^n\) and \(\mathbb C^n\) (so give rise to metric spaces).
Given a finite dimensional real [or complex] vector space \(V\) with a (positive definite) inner product \(\left< \, . \, \right>\), let \[d(v, w): \: = %\:\|v-w \| \: = \: \sqrt{\left<v-w, \: v-w \right>} %\qquad \qquad [=\|v-w \| \,]\] for \(v,w \in V\). Show directly that \(d\) is a metric on \(V\). [Hint: for property (D3) use Cauchy-Schwarz.]
In the space \(C([a,b])\) of continuous functions defined on a closed interval \([a,b]\) (for \(a<b\)), let \[d_1(f,g): = \int_a^b|f(t)-g(t)| \, dt.\] Show that \(d_1\) is a metric on \(C([a,b])\).
Consider the space \(C([a,b])\) of continuous functions on an interval \([a,b]\).
Show that \[d(f,g) := \max_{x \in [a,b]} \vert f(x) - g(x)\vert, \quad f, g \in C([a,b]),\] defines a metric on \(C([a,b])\).
Let \(f (x) = x^2 +7x - 3 \in C([-1,1])\). Describe the open ball \(B_2(f)\) and the closed ball \(\overline{B}_2(f)\).
Let \(S\) be non-empty set. Verify that the standard discrete metric is indeed a metric on \(S\). Hence or otherwise, show that for any \(n \in \mathbb N\) the function \[d_n(\boldsymbol{x},\boldsymbol{y}): = \# \{j: 1 \leq j \leq n \text{ and } x_j \neq y_j\}, \quad \quad (\boldsymbol{x} = (x_1, \ldots, x_n), \boldsymbol{y} = (y_1, \ldots, y_n) \text{ with } x_i, y_i \in S)\] defines a metric on \(S^n:=S \times \cdots \times S\) (\(n\) times). Here, ‘\(\#A\)’ denotes the number of elements in a set \(A\). [When \(S=\{0,1\}\) the metric \(d_2\) is the so-called Hamming metric in communication theory.]
When \(S=\mathbb R\) and \(n=2\), describe the ball \(B_r((0,0))\) in the cases \((a) \: r<1; \:(b) \: r > 2; \: (c) \: 1 \leq r \leq 2. \:\)
For \(\boldsymbol x = (x_1, x_2)\) and \(\boldsymbol y = (y_1, y_2)\) in \(\mathbb R^2\) define the ‘Jungle river’ metric on \(\mathbb R^2\) by \[d(\boldsymbol x, \boldsymbol y): = \begin{cases} |x_2-y_2| \ & \text{ if } \quad x_1=y_1. \\ |x_2|+|y_2| + |x_1-y_1| \ & \text{ if } \quad x_1 \neq y_1. \end{cases}\]
Describe geometrically how \(d\) measures the distance between two points in \(\mathbb R^2\) and verify it is a metric.
Sketch the open balls \(B_1(\boldsymbol{0})\) and \(B_4((3,2))\) in \(\mathbb R^2\) with respect to this metric.
Show that in any metric space \((X,d)\) the set \(\{x\}\), consisting of a single point \(x \in X\), is closed.
Show that in any metric space \((X, d)\) the closed ball \(\overline{B}_r(x):= \{y \in X: d(y,x) \leq r \}\), of radius \(r>0\) centred at \(x \in X\), is closed.
Verify that \(\mathbb H=\{x+iy\in \mathbb C\;|\; x\in\mathbb R, y>0\}\) and \(\mathbb C^\ast=\left\lbrace z\in\mathbb C\;|\; z\not=0\right\rbrace\) and \(\mathbb C\setminus\mathbb R_{\leq 0}\) are open subsets of \(\mathbb C\), but the set \(\mathbb C\setminus\mathbb R_{< 0}\) is neither open nor closed in \(\mathbb C\).
Let \((X,d)\) be a metric space. Show that \(X\) is “Hausdorff"; that is, for any pair of distinct points \(x\) and \(y\) in \(X\) there exist open sets \(U\) and \(V\) such that \(x\) is in \(U\), \(y\) is in \(V\), and \(U \cap V = \emptyset\). (So in metric spaces we can separate points by open sets.)
Let \(A\) be a subset of a metric space \(X\). As in lectures, we define the interior \(A^0\) of \(A\) by \[A^0 :=\{x \in A: \, \text{there exists an open set } U\subseteq A \text{ such that } x \in U \}.\]
Clearly the interior \(A^0\) is open. Show that it is the largest open subset of \(A\); precisely, show that if \(U\) is open and \(U \subseteq A\) then \(U \subseteq A^0\). Deduce that \(A^0\) is the union of all open subsets of \(A\); that is, \[A^0 \: = \: \bigcup_{\substack{U\subseteq A \\ U \text{ open}}} U.\]
Show that for any two subsets \(A, B \subseteq X\) we have \(A^0 \cup B^0 \subseteq (A \cup B)^0\). Write down the interior of the interval \([2,3)\) in \(\mathbb R\). Hence or otherwise, give an example of two subsets of \(\mathbb R\) for which we have \(A^0 \cup B^0 \neq (A \cup B)^0\).
Let \(A\) be a subset of a metric space \(X\). We define the closure \(\overline{A}\) of \(A\) by \[\overline{A} = \{x \in X: U \cap A \ne \emptyset \; \text{for all open sets $U$ containing $x$}\}.\]
Show that \(\overline{A} = \{x \in X: \inf_{z\in A} \: d(x,z)=0\}\).
Show directly from the definition that \(\overline{A}\) is closed.
Consider \(\mathbb R\) together with
the usual metric coming from the absolute value. Show:
(i) The set \(\{x\}\), where \(x\in \mathbb R\) is given, is not
open.
(ii) All open intervals are open, all closed intervals are closed.
(iii) Infinite intersections of open sets are not necessarily
open.
(iv) The interval \((0,1]\) is neither
open nor closed. What is its closure?
Give an example of a metric space \(X\) and an \(x \in X\) such that \(\overline{B}_1(x) \not= \overline{B_1(x)};\) that is, the closure of the open ball is not necessarily the closed ball!!
Let \(A\) be a subset of a metric space \(X\). Show that We may define the boundary \(\partial{A}\) of \(A\) by
\(\partial{A} = \{x \in X: \text{for all open sets $U$ containing $x$, there exist } y,z \in U \text{ with } y \in A \text{ and } z \in A^c\}.\)
a set \(A\) is open if and only if \(\partial A \cap A = \emptyset\);
\(A\) is closed if and only if \(\partial A \subseteq A\).
Show that if a sequence \(\{x_n\}\) converges in a discrete metric space, then it is eventually constant.