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# Metric spaces

### Definition

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.

Ex. • 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 $X$.

Proof

Proof

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 ($-x$).