229 Metric Space
A set with a metric function that is used for calculating the distance between any two elements to study limits and sequences.
Note: Also called distance function for the metric \(d\).
definition [d] (Metric Space) From Kreyszig Definition 1.1-1: a metric space is a pair \((X,d)\), where \(X\) is a set and \(d\) is a metric on \(X\), that is, a function defined on \(X \times X\) such that for all \(x,y,z \in X\) we have:
- (M1) \(d\) is real-valued, finite and nonnegative.
- (M2) \(d(x,y) = 0\) if and only if \(x = y\).
- (M3) \(d(x,y) = d(y,x)\) (Symmetry).
- (M4) \(d(x,y) \leq d(x,z) + d(z,y)\) (Triangle inequality).
where
- \(X\) is a set.
- \(d\) is the metric on \(X\).
- \(x, y, z \in X\).
229.1 Elementary Example
229.1.1 Simple
A metric \(d\) assigns a distance to each pair of points. On three points of the line, use absolute difference.
\[ X = \{ 0,\ 1,\ 2 \} \]
\[ d(x,y) = |x - y| \]
\[ d(0,2) = 2,\quad d(1,1) = 0 \]
where
- \(X\) is the underlying set.
- \(d\) is the metric on \(X\).
229.1.2 General
On \(\mathbb{R}^{3}\) the Euclidean metric is the length of the difference vector.
\[ X = \mathbb{R}^{3} \]
\[ d(x,y) = \sqrt{(x_{1}-y_{1})^{2} + (x_{2}-y_{2})^{2} + (x_{3}-y_{3})^{2}} \]
where
- \(x = (x_{1},x_{2},x_{3})\) and \(y = (y_{1},y_{2},y_{3})\) are points of \(\mathbb{R}^{3}\).
- \(d\) satisfies Kreyszig’s axioms (M1)–(M4).