- In \(\mathbb{R}^n\), define
\[ d'(\mathbf{x}, \mathbf{y}) = \abs{x_1 - y_1} + \dots + \abs{x_n - y_n}. \]
Show that \(d'\) is a metric that induces the usual topology of \(\mathbb{R}^n\). Sketch the basis elements under \(d'\) when \(n = 2\).
- More generally, given \(p \geq 1\), define
\[ d'(\mathbf{x}, \mathbf{y}) = \left[ \sum_{i=1}^{n} \abs{x_i - y_i}^p \right]^{1/p} \]
for \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\). Assume that \(d'\) is a metric. Show that it induces the usual topology on \(\mathbb{R}^n\).
Solution
Let \(\rho(\mathbf x,\mathbf y)=\max_i\abs{x_i-y_i}\) be the square metric, which induces the usual topology of \(\RR^n\), and write \(d_p\) for the metric of (b), so that \(d_1\) is the metric \(d'\) of (a).
1. \(d_1\) is a metric.
\(d_1(\mathbf x,\mathbf y)\ge0\), with equality if and only if \(x_i=y_i\) for every \(i\), and \(d_1\) is symmetric. Summing \(\abs{x_i-z_i}\le\abs{x_i-y_i}+\abs{y_i-z_i}\) over \(i\) gives the triangle inequality.
2. For \(p\ge1\), \(\rho\le d_p\le n^{1/p}\rho\).
For each \(j\), \(\abs{x_j-y_j}=(\abs{x_j-y_j}^p)^{1/p}\le d_p(\mathbf x,\mathbf y)\); take the maximum over \(j\). Also \(d_p(\mathbf x,\mathbf y)\le\bigl(n\,\rho(\mathbf x,\mathbf y)^p\bigr)^{1/p}=n^{1/p}\rho(\mathbf x,\mathbf y)\).
3. For \(p\ge1\), \(d_p\) induces the usual topology of \(\RR^n\).
By step 2, \(B_\rho(\mathbf x,\varepsilon/n^{1/p})\subseteq B_{d_p}(\mathbf x,\varepsilon)\subseteq B_\rho(\mathbf x,\varepsilon)\) for every \(\mathbf x\) and \(\varepsilon>0\). So every \(d_p\)-ball about a point contains a \(\rho\)-ball about it and conversely, and the two metrics have the same open sets.
4. For \(n=2\), the \(d_1\)-ball \(B_{d_1}((x_0,y_0),r)\) is the open square with vertices \((x_0\pm r,y_0)\) and \((x_0,y_0\pm r)\), whose sides make angles of \(45^\circ\) with the axes.
The inequality \(\abs{x-x_0}+\abs{y-y_0}<r\) describes, in each quadrant about \((x_0,y_0)\), the open half-plane bounded by the line through two adjacent vertices.
5. Q.E.D.
Steps 1, 3 with \(p=1\), and 4 answer (a); step 3 answers (b).