Notes on metric space

Metric space (度量空间) is a set with a metric. Symbolically, it is a pair $(X, d)$ where $X$ is the underlying set and $d$ is the metric.

Subspace $(A, d)$ of a metric space $(X, d)$, where $A \subset X$, is also a metric space.

Product space $(X \times Y, d_x \times d_y)$ of two metric spaces $(X, d_x)$ and $(Y, d_y)$, where $d_x \times d_y ((x_1, y_1), (x_2, y_2)) = d_x(x_1, x_2) + d_y(y_1, y_2)$, is also a metric space.

## Metric

Metric specifies the distance among elements within a space. Metric (度量) is a bi-variate function $d(x,y)$ which satisfies:

1. Non-negativity (positivity): $d(x, y) \ge 0, \forall x, y \in X$ ($d(x, y) > 0, \forall x,y \in X, x \ne y$);
2. Triangular inequality: $d(x, y) + d(y, z) \ge d(x, z), \forall x,y,z \in X$;
3. Symmetry: $d(x, y) = d(y, x), \forall x,y \in X$;

A psudo-metric is a function satisfying all the requirements except positivity. For example, for space $C(\mathbb{R},\mathbb{R})$, $$\rho_n(x,y) = \sup_{t\in [-n,n]} |x(t)-y(t)| \quad (n=1,2,3,\cdots)$$ is a psudo-metric. But it can be transformed into a metric, say $$\sigma(x,y) = \sum_{n=1}^{\infty} \frac{1}{2^n} \min {1,\rho_n(x,y) }$$

Distance to a set $d(x, A) = \inf \{ d(x,y): y\in A \}$. Diameter of a set $\mathrm{diam}(A) = \sup \{ d(x,y): x,y\in A \}$. A set is bounded if its diameter is finite.

### $L^p$ metric

L^p metric is a group of metric induced from $L^p$ norm, with $p \in \mathbb{N}_{+}$.

Discrete & finite form (component of point seen as index): $$d_p(x,y) = \left[ \sum_i |x_i - y_i|^p \right]^{\frac{1}{p}}$$ Special cases: $d_1(x,y) = \sum_i |x_i - y_i|$; $d_{\infty}(x,y) = \max_i |x_i - y_i|$.

Continuous & bounded form (component of function seen as index): Take metric space $X = ( C([0,T], \mathbb{R}), d_p )$ as instance, $$d_p(x,y) = \left[ \int_0^T |x(t) - y(t)|^p \text{d} t \right]^{\frac{1}{p}}$$

## Convergence

Given a sequence $\{x_n\}$ in a metric space $(\mathcal{L}, d)$, if $\forall \varepsilon >0, \exists N \in \mathbb{N}$, s.t. $\forall n>N, n\in \mathbb{N}, d(x_n,x) < \varepsilon$, then we say sequence $\{x_n\}$ converges to $x$ (in metric space $\mathcal{L}$ ).

A sequence may converge in one metric, but does not in another.

## Continuous Mapping

continuity, uniform continuity

## Metric-induced Topology

local neighborhoods: open ball, closed ball, sphere

Theorem: (equivalence of metric and topological definitions of continuous mapping)

open set, open mapping

## Completeness

A Cauchy sequence is a sequence $\{a_i\}_{i=1}^N$ in a metric space that satisfies: $\forall \epsilon > 0$, $\exists N \in \mathbb{N}:$, $\forall m, n > N$, $d(a_m, a_n) < \epsilon$.

A metric space is complete if every Cauchy sequence in it converges.

Theorem: (equivalence of metric and topological definitions of complete space)

Baire's Theorem

Contraction mapping theorem

Completion

Equivalent sequence

Theorem: (Completion of metric space)

## Compactness

Sequentially compact.

Boltzano-Weierstras property

Covering, sub-covering, open covering, compact.

The covering number $N(S,s)$ of a metric space $S$ with disks at radius $s$ is the minimal number of such disks needed to cover the space. This number is finite if the metric space is compact.

Compactness Theorem: equivalence of sequentially compact, compact, Boltzano-Weierstras property, H-compact

Theorem: continuous mapping from a compact space induces another compact space.

Theorem: product space of compact spaces is compact.

Boltzano-Weierstras theorem

Theorem: real continuous mapping from a compact space has maximum and minimum.

Pointwise compace, equi-continuous.

Theorem: (Arzela-Ascoli)

In case of function spaces, compact spaces typically are similar to finite-dimensional space.