Norms, metrics, topology

Topological space

A topological space is the fundamental subject of the subdiscipline topology of mathematics. By introducing a topological structure on a set, it is possible to establish.

• intuitive positional relations like "proximity" and
• "convergence against" from the real numbers ${\displaystyle {\mathbb {R} }}$ or from the ${\displaystyle {\mathbb {R} }^{n}}$, respectively.

to many and very general structures (such as the topology of function spaces).

Definition: topology

A topology is a set system ${\displaystyle {\cal {T}}}$ consisting of subsets (open sets) of a basic set ${\displaystyle X}$ for which the following axioms are satisfied.

• (T1) ${\displaystyle \emptyset ,X\in {\cal {T}}}$
• (T2) ${\displaystyle U\cap V\in {\cal {T}}}$ for all ${\displaystyle U,V\in {\cal {T}}}$.
• (T3) For any index set ${\displaystyle I}$ and ${\displaystyle U_{i}\in {\cal {T}}}$ for all ${\displaystyle i\in I}$ holds: ${\displaystyle \bigcup _{i\in I}U_{i}\in {\cal {T}}}$.

A set ${\displaystyle X}$ together with a topology ${\displaystyle {\cal {T}}}$ on ${\displaystyle X}$ is called topological space ${\displaystyle (X,{\cal {T)}}}$.

Definition - open kernel of a set

Let ${\displaystyle M\subseteq X}$ be in the topological space ${\displaystyle (X,{\cal {T)}}}$, then the open core ${\displaystyle {\stackrel {\circ }{M}}}$ is defined as the "largest" open set contained in ${\displaystyle M}$:

${\displaystyle {\stackrel {\circ }{M}}:=\bigcup _{U\in {\mathcal {T}},U\subseteq M}U}$

Definition - connection of a set

Let ${\displaystyle M\subseteq X}$ be in the topological space ${\displaystyle (X,{\cal {T)}}}$, then the closure ${\displaystyle {\overline {M}}}$ is defined as the "smallest" closed set containing ${\displaystyle M}$:

${\displaystyle {\overline {M}}:=\bigcap _{U\in {\mathcal {T}},U^{c}\supseteq M}U^{c}}$

Definition - Boundary of a set

Let ${\displaystyle M\subseteq X}$ be a set in the topological space ${\displaystyle (X,{\cal {T)}}}$, then the edge of a set from the closure of the set ${\displaystyle M}$ without the open core ${\displaystyle {\stackrel {\circ }{M}}}$ of ${\displaystyle M}$. The boundary is therefore defined as follows:

${\displaystyle \partial {M}:={\overline {M}}\setminus {\stackrel {\circ }{M}}}$.

Definition - Neighbourhood

Let ${\displaystyle x_{o}\in X}$ and ${\displaystyle U\subseteq X}$ be a set in a topological space ${\displaystyle (X,{\cal {T)}}}$, then the ${\displaystyle U}$ is called Neighbourhood of ${\displaystyle x_{o}}$ if there exists an open set ${\displaystyle {\stackrel {\circ }{U}}\in {\cal {T}}}$ with:

${\displaystyle x_{o}\in {\stackrel {\circ }{U}}\subseteq U}$

The set of all Neighbourhood of ${\displaystyle x_{o}}$ with respect to the topology ${\displaystyle {\mathcal {T}}}$ is denoted by ${\displaystyle {\mathfrak {U}}_{\mathcal {T}}(x_{o})}$. ${\displaystyle {\stackrel {\circ }{\mathfrak {U}}}_{\mathcal {T}}(x_{o})}$ denotes the set of all open neighbourhoods of ${\displaystyle x_{o}}$.

Definition - Neighbourhood basis

Let ${\displaystyle x_{o}\in X}$ and ${\displaystyle {\cal {{B}\subseteq \wp (X)}}}$ a set system in a topological space ${\displaystyle (X,{\cal {T)}}}$, then the \mathcal{B} is called the neighbourhood basis of ${\displaystyle x_{o}}$ if the following properties hold:

• (B1) ${\displaystyle {\mathcal {B}}\subseteq {\mathfrak {U}}_{\mathcal {T}}(x_{o})}$
• (B2) for each environment ${\displaystyle U\in {\mathfrak {U}}_{\mathcal {T}}(x_{o})}$ a neighborhood ${\displaystyle B\in {\mathcal {B}}}$ with ${\displaystyle B\subseteq U}$.

Definition - base of topology

Let ${\displaystyle x_{o}\in X}$ and ${\displaystyle {\cal {{B}\subseteq \wp {(X)}}}}$ be a set system in a topological space ${\displaystyle (X,{\cal {T)}}}$, then the ${\displaystyle {\mathcal {B}}_{\mathcal {T}}}$ is called the basis of ${\displaystyle {\mathcal {T}}}$ if holds:

• (BT1) ${\displaystyle {\mathcal {B}}_{\mathcal {T}}\subseteq {\mathcal {T}}}$
• (BT2) for every open set ${\displaystyle U\in {\mathcal {T}}}$ there is an open set ${\displaystyle B\in {\mathcal {B}}_{\mathcal {T}}}$ with ${\displaystyle B\subseteq U}$.

Convergence in topological spaces

In calculus, the convergence of sequences is a central definition to define notions based on it, such as continuity, difference, and integrals. Sequences ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ with ${\displaystyle \mathbb {N} }$ as index set are unsuitable to define convergence in general topological spaces, because the index set ${\displaystyle \mathbb {N} }$ is not powerful enough concerning the neighbourhood basis. This is only possible if the topological space has a countable neighbourhood basis. Therefore one goes over either to nets or Filters

Meaning of Properties topology

• (T1) ${\displaystyle \emptyset ,X\in {\mathcal {T}}}$ empty set and the basic set ${\displaystyle X}$ are open sets.
• (T2) ${\displaystyle U\cap V\in {\mathcal {T}}}$ for all ${\displaystyle U,V\in {\mathcal {T}}}$: the average of finitely many open sets is an open set.
• (T3) The union of any many open sets is again an open set.

Semantics: metric

A metric ${\displaystyle d}$ associates with ${\displaystyle d(x,y)}$ two elements ${\displaystyle x,y\in X}$ from a base space ${\displaystyle X}$ the distance ${\displaystyle d(x,y)}$ between ${\displaystyle x}$ and ${\displaystyle y}$.

Definition: Metric

Let ${\displaystyle X}$ be an arbitrary set. A mapping ${\displaystyle d\colon X\times X\to \mathbb {R} }$ is called a metric on ${\displaystyle X}$ if for any elements ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ of ${\displaystyle X}$ the following axioms are satisfied:

• (M1) separation: ${\displaystyle d\left(x,y\right)=0\Leftrightarrow x=y}$,
• (M2) symmetry: ${\displaystyle d\left(x,y\right)=d(y,x)}$,
• (M3) triangle inequality: ${\displaystyle d\left(x,y\right)\leq d(x,z)+d(z,y)}$.

Illustration: metric triangle inequality

According to the triangle inequality, the distance between two points X,Y is at most as large as sum of the distances from X to Z and from Z to Y, that is, a detour via the point Z

Non-negativity

Non-negativity follows from the three properties of the metric, i.e. for all ${\displaystyle x,y\in X}$ holds. ${\displaystyle d(x,y)\geq 0}$. The non-negativity follows from the other properties with:

${\displaystyle 0={\frac {1}{2}}d(x,x)\leq {\frac {1}{2}}(d(x,y)+d(y,x))=}$.

${\displaystyle ={\frac {1}{2}}(d(x,y)+d(y,x))={\frac {1}{2}}(d(x,y)+d(x,y))=d(x,y).}$

Open sets in metric spaces

• In a metric space ${\displaystyle (X,d)}$, one defines a set ${\displaystyle U\subset X}$ to be open (i.e. ${\displaystyle U\in {\mathcal {T}}_{d}}$) if for every ${\displaystyle u\in U}$ there is a ${\displaystyle \epsilon >0}$ that the ${\displaystyle \epsilon }$-sphere ${\displaystyle B_{\epsilon }^{d}(u):=\{\ x\in X|\ d(x,u)<\epsilon \}}$ lies entirely in ${\displaystyle U}$ (i.e. i.e. ${\displaystyle B_{\epsilon }^{d}(u)\subset U}$)
• Show that with this defined ${\displaystyle {\mathcal {T}}_{d}}$, the pair ${\displaystyle (X,{\mathcal {T}}_{d})}$ is a topological space (i.e., (T1), (T2), (T3) satisfied).

Norm on vector spaces

A norm is a mapping ${\displaystyle \|\cdot \|}$ from a vector space ${\displaystyle V}$ over the body ${\displaystyle \mathbb {K} }$ of the real or the complex numbers into the set of nonnegative real numbers ${\displaystyle {\mathbb {R} }_{0}^{+}}$. Here the norm assigns to each vector ${\displaystyle x\in V}$ its length ${\displaystyle \|x\|}$.

Definition: Norm

Let ${\displaystyle V}$ be a ${\displaystyle \mathbb {K} }$ vector space and ${\displaystyle \mathbb {R} _{0}^{+},\;x\mapsto \|x\|,}$ a mapping. If ${\displaystyle \|\cdot \|}$ satisfies the following axioms axioms N1,N2, N3, then ${\displaystyle \|\cdot \|}$ is called a norm on ${\displaystyle V}$.

• (N1) Definiteness: ${\displaystyle \|x\|=0\;\Rightarrow \;x=\mathbb {0} _{V}}$ for all ${\displaystyle x\in V}$,
• (N2) absolute homogeneity: ${\displaystyle \|\lambda \cdot x\|=|\lambda |\cdot \|x\|}$ for all ${\displaystyle x\in V}$ and ${\displaystyle \lambda \in \mathbb {K} }$.
• (N3) Triangle inequality: ${\displaystyle \|x+y\|\leq \|x\|+\|y\|}$ for all ${\displaystyle x,y\in V}$.

Remark: N1

The property (N1) is actually an equivalence and it holds in any normed space. If ${\displaystyle \mathbf {0} _{V}\in V}$ is the zero vector in ${\displaystyle V}$ and ${\displaystyle 0\in \mathbb {K} }$ is the zero in the field ${\displaystyle \mathbb {K} }$, if ${\displaystyle V}$ is a ${\displaystyle \mathbb {K} }$ vector space).

• (N1)' definiteness: ${\displaystyle \|x\|=0\;\Leftrightarrow \;x=\mathbf {0} _{V}}$ for all ${\displaystyle x\in V}$,
• Since one uses a minimality principle for definitions for the defining property, one would not use a stronger formulation (N1)' in the definition for (N1), since the equivalence from the defining properties of the norm follow the properties of the vector space already for any normed space.

${\displaystyle x\equiv \mathbf {0} _{V}\Rightarrow \|x\|=\|0\cdot \mathbf {0} _{V}\|=|0|\cdot \|mathbf{0}_{V}\|=0}$

Normed space / Metric space

A normed space ${\displaystyle (V,\|\cdot \|)}$ is also a metric space.

• A norm ${\displaystyle \|\cdot \|}$ assigns to a vector ${\displaystyle v\in V}$ its vector length ${\displaystyle \|v\|}$.
• The norm ${\displaystyle \|\cdot \|}$ can be used to define a metric via ${\displaystyle d(x,y):=\|x-y\|}$ that specifies the distance between ${\displaystyle x}$ and ${\displaystyle y}$.

Notation: norm

• In the axiom (N2) ${\displaystyle \|\lambda \cdot x\|=|\lambda |\cdot \|x\|}$, ${\displaystyle |\cdot |}$ denotes the amount of the scalar. "${\displaystyle \cdot }$" sign: Outer linkage in vector space or multiplication ${\displaystyle (\mathbb {R} ,\cdot )}$.
• ${\displaystyle \|x\|}$ indicates the length of the vector ${\displaystyle x\in V}$.
• In (N3) ${\displaystyle \|x+y\|\leq \|x\|+\|y\|}$ for all ${\displaystyle x,y\in V}$. '"${\displaystyle +}$"-sign denotes two distinct links (i.e., addition in ${\displaystyle (V,+)}$ and ${\displaystyle (\mathbb {R} ,+)}$, respectively.

Illustration: norm triangle inequality

mini

Def: convergence in normalized space

Let ${\displaystyle (V,\|\cdot \|)}$ be a normalized space and ${\displaystyle (v_{n})_{n\in \mathbb {N} }\in V^{\mathbb {N} }}$ a sequence in ${\displaystyle V}$ and ${\displaystyle v_{o}\in V}$:

${\displaystyle \lim _{n\to \infty }^{\|\cdot \|}v_{n}=v_{o}\$ :\Longleftrightarrow \ \forall _{\epsilon >0}\exists _{n_{\epsilon }\in \mathbb {N} }\forall _{n\geq n_{\epsilon }}\ :\ \|v_{n}-v_{o}\|<\epsilon }

Def: convergence in metric space

Let ${\displaystyle (X,d)}$ be a metric space and ${\displaystyle (x_{n})_{n\in \mathbb {N} }\in X^{\mathbb {N} }}$ a sequence in ${\displaystyle X}$ and ${\displaystyle x_{o}\in X}$:

${\displaystyle \lim _{n\to \infty }^{d}x_{n}=x_{o}\$ :\Longleftrightarrow \ \forall _{\epsilon >0}\exists _{n_{\epsilon }\in \mathbb {N} }\forall _{n\geq n_{\epsilon }}\ :\ d(x_{n},x_{o})<\epsilon }

Def: Cauchy sequences in metric spaces

Let ${\displaystyle (X,d)}$ be a metric space and ${\displaystyle (x_{n})_{n\in \mathbb {N} }\in X^{\mathbb {N} }}$ a sequence in ${\displaystyle X}$. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ is called a Cauchy sequence in ${\displaystyle X^{\mathbb {N} }}$:

${\displaystyle \forall _{\epsilon >0}\exists _{n_{\epsilon }\in \mathbb {N} }\forall _{m,n\geq n_{\epsilon }}\$ :\ d(x_{n},x_{m})<\epsilon }

Equivalence: norms

Let two norms ${\displaystyle \|\cdot \|_{1}}$ and ${\displaystyle \|\cdot \|_{2}}$ be given on the ${\displaystyle \mathbb {K} }$ vector space ${\displaystyle V}$. The two norms are equivalent if holds:

${\displaystyle \exists _{C_{1},C_{2}>0}\forall _{x\in V}\$ :\ C_{1}\|x\|_{1}\leq \|x\|_{2}\leq C_{2}\|x\|_{1}} .

Show that a sequence converges in ${\displaystyle \|\cdot \|_{1}}$ exactly if it also converges with respect to ${\displaystyle \|\cdot \|_{2}}$.

Absolute value in complex numbers

Let ${\displaystyle z=z_{1}+i\cdot z_{2}\in \mathbb {C} }$ be a complex number with ${\displaystyle z_{1},z_{2}\in \mathbb {R} }$. Show that ${\displaystyle |z|:={\sqrt {z\cdot {\overline {z}}}}}$ is a norm on the ${\displaystyle \mathbb {R} }$ vector space ${\displaystyle \mathbb {C} }$!

Historical Notes: Norm

This axiomatic definition of norm was established by Stefan Banach in his 1922 dissertation. The norm symbol in use today was first used by Erhard Schmidt in 1908 as the distance ${\displaystyle \|x-y\|}$ between vectors ${\displaystyle x}$ and ${\displaystyle y}$.

See also

• Absolute value
• Complex Analysis
• Inverse-producing extensions of Topological Algebras
• Functional analysis

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