Notes on metric space

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

subspace,

product space $(X\times Y, d_x \times d_y )$

Metric

Metric specifies the distance among elements within a space.

Bi-variate function $d(x,y)$ is a metric if it satisfies:

1. Non-negativity (positivity)
2. Triangular inequality
3. Symmetry

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 $p \in \mathbb{N}_{+}$.

1. Discrete & finite form: (component of point seen as index) $$d_p(x,y) = \left[ \sum_i |x_i - y_i|^p \right]^{\frac{1}{p}}$$
• $d_1(x,y) = \sum_i |x_i - y_i|$
• $d_{\infty}(x,y) = \max_i |x_i - y_i|$
2. Continuous & bounded form: (component of function seen as index)
• Take metric space $X = ( C([0,T], \mathbb{R}), d_p )$ as instance, $$d_p(x,y) = \left[ \int_0^T |x(t) - y(t)|^p \text{d} t \right]^{\frac{1}{p}}$$

Psudo-metric

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 $$\sigma(x,y) = \sum_{n=1}^{\infty} \frac{1}{2^n} \min {1,\rho_n(x,y) }$$

Convergence

Given $\{x_n\}, x \in (\mathcal{L}, d(\cdot,\cdot) )$, 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 doesnot in another.

Continuous Mapping

continuity, uniform continuity

Metric-induced Topology

local neighborhoods: open ball, closed ball, sphere

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

open set, open mapping

Completeness

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

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

Baire's Thm

Contraction mapping thm

Completion

equivalent sequence

Thm: (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 Thm: equivalence of sequentially compact, compact, Boltzano-Weierstras property, H-compact

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

Thm: product space of compact spaces is compact.

Boltzano-Weierstras theorem

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

pointwise compace, equi-continuous

Thm: (Arzela-Ascoli)

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