# Metric Spaces

Definition (Metric, Metric Space)

A

metric(ordistance function) on a set $X$ is a mapping $d:X×X→R$ with the properties

(M1)$∀x,y∈X:d(x,y)=0⟺x=y$(point separation)

(M2)$∀x,y∈X:d(x,y)=d(y,x)$(symmetry)

(M3)$∀x,y,z∈X:d(x,z)≤d(x,y)+d(y,x)$(triangle inequality)

We say that $d(x,y)$ is thedistancebetween $x$ and $y$.

Ametric spaceis a pair $(X,d)$ consisting of a set $X$ and a metric $d$ on $X$.

Setting $x=z$ in **(M3)** and applying **(M1)** & **(M2)** gives us $0=d(x,x)≤2d(x,y)$, hence $d(x,y)≥0$. This *nonnegativity* of the metric is often part of the definition.

Relaxing **(M1)** to the condition $∀x∈X:d(x,x)=0$ leads to the notion of a *semi-metric* and that of a *semi-metric space*. Nonnegativity still follows as shown above.

*Pseudo-metric* is usually a synonym for *semi-metric*.

*Quasi-metric* refers to dropping **(M2)**.

An *ultrametric* satisfies in place of **(M3)** the stronger condition $d(x,z)≤max{d(x,y),d(y,z)}$.

Definition (Metric Subspace)

A

metric subspaceof a metric space $(X,d)$ is a pair $(S,d_{S})$ where $S$ is a subset of $X$ and $d_{S}$ is the restriction of $d$ to $S×S$.

Clearly, a metric subspace of a metric space is itself a metric space.

Let $(X,d)$ be a (semi-)metric space.

For all $x,y,z∈X$ we have the

$∣d(x,y)−d(y,z)∣≤d(x,z).$inverse triangle inequalityFor all $v,w,x,y∈X$ we have the

$∣d(v,w)−d(x,y)∣≤d(v,x)+d(w,y)$quadrilateral inequality

The proofs are straightforward.

Definition (Isometry)

Suppose $(X,d_{X})$ and $(Y,d_{Y})$ are metric spaces. We say that a mapping $f:X→Y$ is

isometricor anisometryif it obeys $d_{Y}(f(x),f(x_{′}))=d_{X}(x,x_{′})$ for all $x,x_{′}∈X$.

As a consequence of **(M1)**, every isometry is injective.

TODO

- metric induced by a norm
- metric product

Definition (Diameter)

The

$diam(S)=sup{d(x,y):x,y∈S}∈{−∞}∪[0,∞].$diameterof a subset $S$ of a metric space $(X,d)$ is the number

Note that $diam(S)=−∞$ iff $S=∅$, and $diam(S)=0$ iff $S$ is a singleton set.

Definition (Distance)

Suppose $(X,d)$ is a metric space. The

$dist(x,S)=f{d(x,y):y∈S}∈[0,∞].$distancefrom a point $x∈X$ to a subset $S⊂X$ is

Note that $dist(x,S)=∞$ iff $S=∅$.

Definition (Convergence, Limit, Divergence)

Let $(X,d)$ be a metric space. A sequence $(x_{n})_{n∈N}$ in $X$ is said to

$∀ϵ>0∃N∈N∀n≥N:d(x,x_{n})<ϵ.$converge to a point $x∈X$, ifIn this case, we call $x$ a

$n→∞lim x_{n}=x$limit (point)of the sequence. Symbolically this is expressed byor by saying that $x_{n}→x$ as $n→∞$.

We call a sequence in $X$

convergentif it converges to some point of $X$ anddivergentotherwise.

For a semi-metric space the definition remains the same. However, the notation $limx_{n}=x$ can be misleading, because there might be more than one limit point.

A sequence in a metric space has at most one limit.

In other words: The limit of a convergent sequence in a metric space is unique.

Proof Let $(x_{n})$ be a convergent sequence in a metric space $(X,d)$ with limit point $x$. If $x_{′}$ is another limit point of $(x_{n})$, then $d(x,x_{′})≤d(x,x_{n})+d(x_{n},x_{′})$ for all $n∈N$ by **(M3)**. Given $ϵ>0$, there exist natural numbers $N$ and $N_{′}$ such that $d(x,x_{n})<ϵ$ for all $n≥N$ and $d(x,x_{n})<ϵ$ for all $n≥N_{′}$. Both hold, if $n$ is large enough ($≥max{N,N_{′}}$ to be precise). It follows that $d(x,x_{′})<2ϵ$. Since $ϵ$ was arbitrary, $d(x,x_{′})=0$. Therefore, $x=x_{′}$ by **(M1)**. $□$

A semi-metric space $X$ is a metric space if and only if every sequence in $X$ has at most one limit.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. A mapping $f:X→Y$ is called

$∀ϵ>0∃δ>0∀x_{′}∈X:(d_{X}(x,x_{′})≤δ⟹d_{Y}(f(x),f(x_{′}))<ϵ)$

continuous at a point $x∈X$if$∀x∈X∀ϵ>0∃δ>0∀x_{′}∈X:(d_{X}(x,x_{′})≤δ⟹d_{Y}(f(x),f(x_{′}))<ϵ)$

continuousif it is continuous at every point of $X$, that is$∀ϵ>0∃δ>0∀x,x_{′}∈X:(d_{X}(x,x_{′})≤δ⟹d_{Y}(f(x),f(x_{′}))<ϵ)$

uniformly continuousif$∃L≥0∀x,x_{′}∈X:d_{Y}(f(x),f(x_{′}))≤Ld_{X}(x,x_{′})$

Lipschitz continuousif

Definition (Open Ball, Closed Ball, Sphere)

Suppose $(X,d)$ is a metric space. The

$B(c,r)={x∈X:d(x,c)<r}.$open ballwith center $c∈X$ and radius $r>0$ is the setThe

$B(c,r)={x∈X:d(x,c)≤r}.$closed ballwith center $c∈X$ and radius $r>0$ is the setThe

$S(c,r)={x∈X:d(x,c)=r}.$spherewith center $c∈X$ and radius $r>0$ is the set

Observe that $S(c,r)=B(c,r)∖B(c,r)$.

Definition (Open Subset of a Metric Space)

A subset $O$ of a metric space is called

openif for every point $x∈O$ there exists an $ϵ>0$ such that $B(x,ϵ)⊂O$.

Proposition (Metric Topology)

Let $(X,d)$ be a metric space. The collection of open subsets of $X$ forms a topology on $X$. This topology is called the

metric topologyon $X$ induced by $d$.

- Open balls in a metric space are open with respect to the metric topology.
- Closed balls in a metric space are closed with respect to the metric topology.
- The boundary (with respect to the metric topology) of an open or closed ball is the sphere with the same center and radius. Not true!!!!
- The collection of open balls in a metric space forms a basis of the metric topology.

## Complete Metric Spaces

- Definition
- Banach Fixed-Point Theorem
- Baire
- Metric Completion