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Metric spaces


Let \(X\) be a set and \(d \colon X \times X \to [0,\infty)\) a function such that \[ \begin{align*} &\mathrm{(i)} \qquad &d(x,y) &= d(y,x), \qquad&\text{(symmetry)}&\\ &\mathrm{(ii)} \qquad &d(x,y) &\leq d(x,z) + d(z,y), \qquad&\text{(triangle inequality)}&\\ &\mathrm{(iii)} \qquad &d(x, y) &= 0 \quad\text{ if only if }\quad x = y. \qquad&\text{(non-degeneracy)}&\\ \end{align*} \] Then the pair \((X,d)\) is called a metric space and the function \(d\) is called a metric or distance on \(X\).

A subset \(M \subset X\) is called a subspace of \(X\), written \((M,d) \subset (X,d)\), if \(M\) is endowed with the same metric as \(X\), called the induced metric on \(M\). Subspaces of metric spaces are themselves metric spaces.

  • Any set becomes a metric space when endowed with the discrete metric \[ d(x,y) := \begin{cases} 1, \qquad &x \neq y,\\ 0, \qquad &x = y. \end{cases} \]

  • \(\R^n\) becomes a metric space when endowed with the Euclidean distance \[ d(x,y) := |x - y| = \big( (x_1-y_1)^2 +\ldots + (x_n-y_n)^2\big)^{1/2}.\]
  • The supremum norm induces a metric on the set of bounded and continuous functions on an interval: \[ d(f,g) = \|f-g\|_\infty = \sup_{x \in I}|f(x)-g(x)|. \] This makes \((BC(I,\mathbb R), \| \cdot \|_\infty)\) a metric space.

℘ Normed spaces are metric spaces

If \(\|\cdot\|\) is a norm on \(X\), then \(d(x,y) := \|x-y\|\) is a metric on \(X\).



The distance is non-negative and well defined, since \[ 0 \leq \underbrace{\|x-y\|}_{d(x,y)} \leq \|x\| + \|y\| < \infty, \quad\text{ for } \quad x,y \in (X, \|\cdot\|). \] \[ \begin{align*} &\text{Symmetry:} \qquad &d(x,y) &= \|x-y\| = \|y-x\| = d(y,x).\\ &\text{Triangle inequality:} \qquad &d(x,y) &= \|x-y\| \leq \|x-z\| + \|z-y\| = d(x,z) + d(z,y).\\ &\text{Non-degeneracy:} \qquad &d(x,y) &= \|x-y\| = 0 \quad \Longleftrightarrow \quad x - y = 0 \quad \Longleftrightarrow \quad x = y. \end{align*} \]

N.b. Metric spaces need not be vector spaces. The set of positive real numbers, \(\R_+ = (0,\infty)\), with the metric given by \(d(x,y) := |x-y|\) is a metric space, but it is not a linear space, since it contains neither an additive identity (\(0\)) nor additive inverses (\(-x\)).

2017-03-24, Hallvard Norheim Bø