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Section Summary
Important ideas that we discussed in this section include the following.
A metric on a space \(X\) is a function that measures distance between elements in the space. More formally, a metric on a space \(X\) is a function \(d: X \times X \to \R^+ \cup \{0\}\) such that
\(d(x,y) \geq 0\) for all \(x,y \in X\text{,}\)
\(d(x,y) = 0\) if and only if \(x = y\) in \(X\text{,}\)
\(d(x,y) = d(y,x)\) for all \(x, y \in X\text{,}\) and
\(d(x,y) \leq d(x,z) + d(z,y)\) for all \(x,y,z \in X\text{.}\)
A metric space is any space combined with a metric defined on that space.
The Euclidean, taxicab, and max metric are all metrics on \(\R^n\text{,}\) so they all provide ways to measure distances between points in \(\R^n\text{.}\) These metric are different in how they define the distances.
The Euclidean metric is the standard metric that we have used through our mathematical careers. For elements \(x = (x_1, x_2, \ldots,
x_n)\) and \(y = (y_1, y_2, \ldots,
y_n)\) in \(\R^n\text{,}\) the Euclidean metric \(d_E\) is defined as
\begin{align*}
d_E(x,y) \amp = \sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + \cdots (x_n-y_n)^2}\\
\amp = \sqrt{\sum_{i=1}^n (x_i-y_i)^2}\text{.}
\end{align*}
With this metric, the unit circle in \(\R^2\) (the set of points a distance \(1\) from the origin) is the standard unit circle we know from Euclidean geometry.
The taxicab metric \(d_T\) is defined as
\begin{align*}
d_T(x,y) \amp = |x_1-y_1| + |x_2-y_2| + \cdots + |x_n-y_n|\\
\amp = \sum_{i=1}^n |x_i-y_i|\text{.}
\end{align*}
The unit circle in \(\R^2\) using the taxicab metric is the square with vertices \((1,0)\text{,}\) \((0,1)\text{,}\) \((-1,0)\text{,}\) and \((0,-1)\) when viewed in Euclidean geometry.
The max metric \(d_M\) is defined by
\begin{align*}
d_M(x,y) \amp = \max\{| x_1-y_1 |, | x_2-y_2 |, |x_3-y_3|, \ldots, |x_n-y_n| \}\\
\amp = \max_{1 \leq i \leq n} \{|x_i-y_i|\}\text{.}
\end{align*}
Under the max metric, the unit circle in \(\R^2\) is the square with vertices \((1,1)\text{,}\) \((-1,1)\text{,}\) \((-1,-1)\text{,}\) and \((1,-1)\) when viewed in Euclidean geometry.
