**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 specifies the **distance** among elements within a space.

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

- Non-negativity (positivity)
- Triangular inequality
- 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** is a group of metric induced from $L^p$ norm, with $p \in \mathbb{N}_{+}$.

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

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

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

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.

continuity, uniform continuity

local neighborhoods: open ball, closed ball, sphere

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

open set, open mapping

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

equivalent sequence

Thm: (Completion of metric space)

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.