Inverse-producing extensions of Topological Algebras/Gauge functionals

Introduction

In real numbers, there is the amount to be able to express convergence in space, for example. With the amount you can define ${\textstyle \varepsilon }$-neighbourhoods and the sequence convergence is defined via these ${\textstyle \varepsilon }$-neighbourhoods. In addition, Minkowski functionals are defined into circular zero environments ${\textstyle U}$, depending on topological properties, the set provides certain properties of the Minkowski functionals.

Convergence in real numbers

The real numbers with the amount ${\textstyle (\mathbb {R} ,|\cdot |)}$ is a normaled space and $(v_{n})_{n\in \mathbb {N} }\in \mathbb {R} ^{\mathbb {N} }$ is a sequence in ${\textstyle \mathbb {R} }$ and $v_{o}\in \mathbb {R}$:

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

Convergence in normed spaces

The convergence in normed spaces ${\textstyle (V,\|\cdot \|)}$ ${\textstyle (v_{n})_{n\in \mathbb {N} }\in V^{\mathbb {N} }}$ is defined analogously as a sequence in ${\textstyle V}$ and $v_{o}\in V$:

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

Epsilon environments

The absolute value or (in more a more general case) the norm can be used in ${\textstyle (V,\|\cdot \|)}$ for the definition of the ${\textstyle \varepsilon }$ neighbourhoods.

$B_{\varepsilon }^{\|\cdot \|}(a):=\left\{x\in V\,:\,\|x-a\|<\varepsilon \right\}$

These topology-producing functionals (gauge functionals) are required for the definition of algebra extensions in which a given ${\textstyle z}$ has an inverse element. The topologising of the algebra of power series is later performed with measuring functionals (e.g. seminorms, ${\textstyle p}$-seminorms, ...)

Absorbent sets

The measurements functional are defined via circularly absorbing zero environments for which the associated Minkowski functional produces the associated measuring functionally. The basics provide the following sections.

Introduction measuring functional

When measuring functionals are used, the defining properties of a standard are further generalized in order to be able to use topology-generating functionals in arbitrary topological algebras in an analogous manner. As a result, it will no longer be necessary to describe, for example, stiffness over the open sets from the topology (see also Theorem of continuity für lineare mappings).

Definition: p-homogeneous

Let ${\textstyle V}$ a vector space over the field ${\textstyle \mathbb {K} }$. A functional $f:V\longrightarrow \mathbb {K}$ is called ${\textstyle p}$-homogeneous, if there is a ${\textstyle p\in \mathbb {K} }$ with ${\textstyle 0<p\leq 1}$, which applies:

$$\forall _{\displaystyle x\in V,\lambda \in \mathbb {K} }:f(\lambda \cdot x)=|\lambda |^{p}\cdot f(x)$$

If ${\textstyle p=1}$, then ${\textstyle f}$ is homogeneous. ${\textstyle f}$ is called non-negative if for all ${\textstyle x\in V}$ applies ${\textstyle f(x)\geq 0}$.

Definition: p-gauge functional

Let${\textstyle V}$ a vector space over ${\textstyle \mathbb {K} }$. A non-negative, ${\textstyle p}$-homogeneous functional ${\textstyle f:V\longrightarrow \mathbb {K} }$ is called ${\textstyle p}$-gauge functional on ${\textstyle V}$ and just gauge functional for ${\textstyle p=1}$.

Example: p-gauge functional

Let${\textstyle 0<p\leq 1}$ and $V_{p}:=\left\{(x_{n})_{n\in \mathbb {N} }\in \mathbb {R} ^{\mathbb {N} }\,:\,\sum _{n=1}^{\infty }|x_{n}|^{p}<\infty {\mbox{ mit }}x=(x_{n})_{n\in \mathbb {N} }\right\}$, then $\|x\|_{p}:=\sum _{n=1}^{\infty }|x_{n}|^{p}$ is a $p$-gauge functional on ${\textstyle V}$.

Learning Task: p-gauge functional

The vector space ${\textstyle V_{p}}$ and the ${\textstyle p}$-gauge functional $\|x\|_{p}:=\sum _{n=1}^{\infty }|x_{n}|^{p}$ are given. Proof that ${\textstyle \left({\frac {1}{n^{2}}}\right)_{n\in \mathbb {N} }\in V_{1}}$ but ${\textstyle \left({\frac {1}{n^{2}}}\right)_{n\in \mathbb {N} }\notin V_{\frac {1}{2}}}$. What is the general definition for ${\textstyle V_{p}}$ and ${\textstyle V_{q}}$ with ${\textstyle p<q}$ and ${\textstyle p,q\in (0,1]}$?

Note

The ${\textstyle p}$ homogeneity has on the one hand a close relationship with the continuity of the scalar multiplication and the ${\textstyle p\in (0,1]}$ defines the relationship with a quasi-seminorm that induces the same open sets on the corresponding $p$-seminorm. The triangle inequation for the $p$-homogeneous $p$-seminorm has a quasi-seminorm with a corresponding concavity constant $K$ with:

$\|x+y\|\leq K\cdot (\|x\|+\|y\|)$

Definition: System of p-gauge functionals

Let${\textstyle V}$ is a vector space over ${\textstyle \mathbb {K} }$, ${\textstyle I}$ an index set and for all ${\textstyle \alpha \in I}$ let ${\textstyle \left\|\cdot \right\|_{\alpha }}$ be a ${\textstyle p}$-gauge functional on ${\textstyle V}$. Then ${\textstyle \left\|\cdot \right\|_{I}}$ denotes the set of all ${\textstyle p}$-gauge functionals with indices of ${\textstyle \alpha \in I}$, i.e.

$$\left\|\cdot \right\|_{I}:=\left\{\left\|\cdot \right\|_{\alpha }:\alpha \in I\right\}.$$

${\textstyle \left\|\cdot \right\|_{I}}$ is called system of ${\textstyle p}$-gauge functionals. If ${\textstyle p=1}$ is called ${\textstyle \left\|\cdot \right\|_{I}}$ strain functionalystem.

Definition: equivalence of p-gauge functionalystemen

Let ${\textstyle V}$ a vector space over ${\textstyle \mathbb {K} }$ and ${\textstyle \left\|\cdot \right\|_{I_{1}}}$, ${\textstyle \left\|\cdot \right\|_{I_{2}}}$ two systems of $p$-gauge functionals on $V$. The system ${\textstyle \left\|\cdot \right\|_{I_{1}}}$ of ${\textstyle p}$-gauge functionals and the system ${\textstyle \left\|\cdot \right\|_{I_{2}}}$ are named as equivalent if the following two conditions are true:

• (EQ1) ${\textstyle \forall _{\alpha \in I_{1}}\exists _{\beta \in I_{2},C_{\beta }>0}\forall _{v\in V}:\,\,\|v\|_{\alpha }\leq C_{\beta }\cdot \|v\|_{\beta }}$
• (EQ2) ${\textstyle \forall _{\beta \in I_{2}}\exists _{\alpha \in I_{1},C_{\alpha }>0}\forall _{v\in V}:\,\,\|v\|_{\beta }\leq C_{\alpha }\cdot \|v\|_{\alpha }}$

Example: p-Gauge functional

With ${\textstyle p>0}$ let ${\textstyle V:={\mathcal {C}}(\mathbb {R} ,\mathbb {R} )}$ be the set of all continuous function on the domain ${\textstyle \mathbb {R} }$ mapping to the set ${\textstyle \mathbb {R} }$. The set of the ${\textstyle p}$-gauge functional is defined as follows:

$\left\|\cdot \right\|_{I}:=\left\{\left\|\cdot \right\|_{\alpha }:\alpha \in I:=\mathbb {N} \right\}.$

and with a specific $\alpha \in I=\mathbb {N}$ the $p$-gauge functional is defined as

$\left\|f\right\|_{\alpha }:=\int _{-\alpha }^{+\alpha }\left|f(x)\right|^{p}d\,x.$

Definition: Base of topology for a p-gauge functionalystem

Let ${\textstyle (V,{\mathcal {T}})}$ a topological vector space with the system of open sets ${\textstyle {\mathcal {T}}}$ on $V$. Furthermore, ${\textstyle \left\|\cdot \right\|_{I}}$ is a set of ${\textstyle p}$-gauge functionals on ${\textstyle V}$. The ${\textstyle p}$-gauge functionalystem is called 'basic' for ${\textstyle {\mathcal {T}}}$ if:

• (BE1) ${\textstyle \forall _{x_{o}\in V}\forall _{\alpha \in I}\forall _{\varepsilon >0}:\,B_{\varepsilon }^{\alpha }(x_{o}):=\left\{x\in V\,:\,\left\|x-x_{o}\right\|_{\alpha }<\varepsilon \right\}\in {\mathcal {T}}}$
• (BE2) ${\textstyle {\forall }_{U\in {\mathcal {T}}}\forall _{x_{o}\in U}\exists _{\alpha \in I,\varepsilon >0}:\,B_{\varepsilon }^{\alpha }(x_{o}):=\left\{x\in V\,:\,\left\|x-x_{o}\right\|_{\alpha }<\varepsilon \right\}\subseteq U}$

Note: Base of topology for a system of p-gauge functionals

• (BE1) means that the ${\textstyle \varepsilon }$ balls ${\textstyle B_{\varepsilon }^{\alpha }(v_{o})}$ are self-open sets.
• With (BE2), each open set ${\textstyle U}$ can be represented from the topology ${\textstyle {\mathcal {T}}}$ as a union of ${\textstyle \varepsilon }$ balls. Since any associations of open sets in a topological space after Axiom (T3) must also be open again, the association of ${\textstyle \varepsilon }$ balls ${\textstyle B_{\varepsilon }^{\alpha }(v_{o})}$ with ${\textstyle \alpha \in I}$, (698-1047-1747220434181.

Definition: Sub-based p-gauge functionalystem

Let${\textstyle (V,{\mathcal {T}})}$ a topological vector space with the system of open sets ${\textstyle {\mathcal {T}}}$ on ${\textstyle V}$. Furthermore, ${\textstyle \left\|\cdot \right\|_{I}}$ is a set of ${\textstyle p}$-gauge functionals on ${\textstyle V}$. The ${\textstyle p}$-gauge functionalystem is called 'sub-based' for ${\textstyle {\mathcal {T}}}$, if applicable with ${\textstyle \,B_{\varepsilon }^{\alpha }(v_{o}):=\left\{v\in V\,:\,\left\|x-x_{o}\right\|_{\alpha }<\varepsilon \right\}}$:

• (SE1) ${\textstyle \forall _{v_{o}\in V}\forall _{\alpha \in I}\forall _{\varepsilon >0}:\,\,B_{\varepsilon }^{\alpha }(v_{o})\in {\mathcal {T}}}$
• (SE2) ${\textstyle {\forall }_{U\in {\mathcal {T}}}\forall _{v_{o}\in U}\exists _{n\in \mathbb {N} }\exists _{\alpha _{1},\ldots ,\alpha _{n}\in I,\varepsilon >0}:\,B_{\varepsilon }^{(\alpha _{1},\ldots ,\alpha _{n})}(v_{o})\subseteq U}$

with

$$B_{\varepsilon }^{(\alpha _{1},\ldots ,\alpha _{n})}(v_{o}):=\left\{v\in V\,:\,\left\|x-x_{o}\right\|_{\alpha _{k}}<\varepsilon {\mbox{ für alle }}k\in \{1,\ldots ,n\}\right\}$$

Note: Difference in topology - sub-based production

In a topology-producing ${\textstyle p}$-gauge functionalystem, the manual handling of finite cuts of open sets in a topology is simplified. (S2) must therefore take into account finite cuts of the neighbourhoods by cutting ${\textstyle \varepsilon }$ balls ${\textstyle B_{\varepsilon _{1}}^{\alpha _{1}},\ldots ,B_{\varepsilon _{n}}^{\alpha _{n}}}$ by the condition

$$\left\|x-x_{o}\right\|_{\alpha _{k}}<\varepsilon _{k}{\mbox{ für alle }}k\in \{1,\ldots ,n\}$$

with ${\textstyle \varepsilon :=\min\{\varepsilon _{k}>0\,:\,k\in \,\{1,\ldots ,n\}\,\}}$.

Definition: unital positive

Let${\textstyle (A,\left\|\cdot \right\|_{I})}$ a unital topological algebra over ${\textstyle \mathbb {K} }$ with the single element of multiplication ${\textstyle e_{A}\in A}$. The ${\textstyle p}$-gauge functionalystem ${\textstyle \left\|\cdot \right\|_{I}}$ is called 'unital positive' exactly when for all ${\textstyle \alpha \in I}$ the condition (698-1047-1747220434181-341150).

Note: unital positive equivalent measuring functionalystem

One can replace a ${\textstyle p}$-gauge functionalystem on a topological algebra with an equivalent unital positive ${\textstyle p}$-gauge functionalystem by using the separation property of a house village woman to use minkowki functionals of circular zero environments which do not contain the single element. Then you get ${\textstyle \left\|e_{A}\right\|_{\alpha }\geq 1}$ when ${\textstyle e_{A}\notin U_{\alpha }}$ and ${\textstyle \left\|x\right\|_{\alpha }:=p_{U_{\alpha }}(x)}$ are used as Minkowski-Funktional of the absorbent zero environment ${\textstyle U_{\alpha }}$.

Note: p-standard and standard

The term standard is a special case of a ${\textstyle p}$ standard with ${\textstyle p=1}$ which is defined below.

Definition: Standard

Let${\textstyle V}$ a topological vector space above the field ${\textstyle \mathbb {K} }$. A functional ${\textstyle \left\|\cdot \right\|:V\longrightarrow \mathbb {K} }$ is called standard on ${\textstyle V}$ if ${\textstyle \left\|\cdot \right\|}$ meets the following conditions:

• (N1) ${\textstyle \displaystyle \forall _{\displaystyle x\in V}:\left\|x\right\|\geq 0}$
• (N2) ${\textstyle \displaystyle \left\|x\right\|=0\Longrightarrow x=0}$

(N3) ${\textstyle \displaystyle \forall _{\displaystyle x\in V,\lambda \in \mathbb {K} }:\left\|\lambda \cdot x\right\|=|\lambda |\cdot \left\|x\right\|}$

• (N4) ${\textstyle \displaystyle \forall _{\displaystyle x,y\in V}:\left\|x+y\right\|\leq \left\|x\right\|+\left\|y\right\|}$

Definition:

Let${\textstyle V}$ a topological vector space above the field ${\textstyle \mathbb {K} }$. A functional ${\textstyle \left\|\cdot \right\|:V\longrightarrow \mathbb {K} }$ is called semistandard on ${\textstyle V}$, if ${\textstyle \left\|\cdot \right\|}$ meets the following conditions:

• (H1) ${\textstyle \displaystyle \forall _{\displaystyle x\in V}:\left\|x\right\|\geq 0}$
• (H2) ${\textstyle \displaystyle \forall _{\displaystyle x\in V,\lambda \in \mathbb {K} }:\left\|\lambda \cdot x\right\|=|\lambda |\cdot \left\|x\right\|}$
• (H3) ${\textstyle \displaystyle \forall _{\displaystyle x,y\in V}:\left\|x+y\right\|\leq \left\|x\right\|+\left\|y\right\|}$

Note:standard semi - standard

If (N2) does not apply in the definition of the standard, a ${\textstyle \left\|\cdot \right\|}$ is obtained semi-standard. (N2) ensures the house village property in the topological vector space. It is possible to separate the points with the standard, i.e. to measure whether two vectors ${\textstyle v_{1},v_{2}\in V}$ differ, i.e. ${\textstyle v_{1}\not =v_{2}}$ or ${\textstyle v_{1}-v_{2}\not =0_{V}}$.

Multiplicative convex - submultiplicity of seminorm

A seminorm is submultiplicative with a stiffness constant ${\textstyle C>0}$, if applicable for all ${\textstyle v_{1},v_{2}\in V}$:

$$\left\|v_{1}\cdot v_{2}\right\|\leq C\cdot \left\|v_{1}\right\|\cdot \left\|v_{2}\right\|$$

${\textstyle C}$ is called the stiffness constant of the multiplication. The seminorm ${\textstyle \left\|\cdot \right\|}$ can be replaced by an equivalent seminorm ${\textstyle \left\|\cdot \right\|_{1}}$ for which ${\textstyle C=1}$ is (see MLC-Regularität.

Lemma: Stability constant and submultiplicity

Let${\textstyle (A,\|\cdot \|_{\mathcal {A}})}$ a local-convexe topological algebra with the basic-generating seminorm system ${\textstyle \|\cdot \|_{\mathcal {A}}}$ and a submultiplicative seminorm with stiffness constant ${\textstyle C>0}$ and ${\textstyle v_{1},v_{2}\in V}$ given with: (698-863-174722043418-341-11) then there is an equivalent seminorm ${\textstyle \|\cdot \|_{\beta }}$ with

$$\left\|v_{1}\cdot v_{2}\right\|_{\beta }\leq \left\|v_{1}\right\|_{\beta }\cdot \left\|v_{2}\right\|_{\beta },$$

Proof: Stability constant and submultiplicity

If ${\textstyle 0<C\leq 1}$ submultiplicity is obtained directly with

$$\left\|v_{1}\cdot v_{2}\right\|_{\alpha }\leq C\cdot \left\|v_{1}\right\|_{\alpha }\cdot \left\|v_{2}\right\|_{\alpha }\leq \left\|v_{1}\right\|_{\alpha }\cdot \left\|v_{2}\right\|_{\alpha }$$

Proof: Definition of seminorm

If ${\textstyle C>1}$ is applicable, this is defined for all ${\textstyle v\in A}$:

$$\left\|v\right\|_{\beta }:=C\cdot \left\|v\right\|_{\alpha }$$

and the submultiplicity is obtained via:

$$\left\|v_{1}\cdot v_{2}\right\|_{\beta }\leq C\cdot \left\|v_{1}\cdot v_{2}\right\|_{\alpha }\leq C^{2}\cdot \left\|v_{2}\right\|_{\alpha }\cdot \left\|v_{2}\right\|_{\alpha }=\left\|v_{1}\right\|_{\beta }\cdot \left\|v_{2}\right\|_{\beta }$$

Proof: equivalence of seminorms

The equivalence of the seminorms is obtained directly from the definition with ${\textstyle C>1}$, for:

$$C\cdot \left\|v\right\|_{\alpha }=\left\|v\right\|_{\beta }$$

Note: Submultiplicity

If a topological algebra is a normed space, it is generally only possible to say that the submultiplicivity of the seminorm is met with a certain stiffness constant of the multiplication, since the ${\textstyle \varepsilon }$ balls around the zero vector generate a neighborhood system of the zero vector. The Lemma shows that without restriction a seminorm with stiffness constant can also be replaced by an equivalent submultiplicative seminorm. The procedure can be carried out analogously for local spaces.

Definition: p-Norm

Let${\textstyle V}$ a vector space above the field ${\textstyle \mathbb {K} }$ and ${\textstyle 0<p\leq 1}$. A functional ${\textstyle \left\|\cdot \right\|:V\longrightarrow \mathbb {K} }$ means ${\textstyle p}$ standard on ${\textstyle V}$, if ${\textstyle \left\|\cdot \right\|}$ meets the following conditions:

• (PN1) ${\textstyle \displaystyle \forall _{\displaystyle x\in V}:\left\|x\right\|\geq 0}$
• (PN2) ${\textstyle \displaystyle \left\|x\right\|=0\Longrightarrow x=0}$
• (PN3) ${\textstyle \displaystyle \forall _{\displaystyle x\in V,\lambda \in \mathbb {K} }:\left\|\lambda \cdot x\right\|=|\lambda |^{p}\cdot \left\|x\right\|}$
• (PN4) ${\textstyle \displaystyle \forall _{\displaystyle x,y\in V}:\left\|x+y\right\|\leq \left\|x\right\|+\left\|y\right\|}$

Note

For ${\textstyle p\geq 1}$, a ${\textstyle p}$ standard can also be made to a standard by setting the standard ${\textstyle \left\|\cdot \right\|_{\ast }}$ as follows:

$$\left\|x\right\|_{\ast }:=\left\|x\right\|^{\frac {1}{p}}$$

Example

Let${\textstyle p\in \mathbb {R} }$ with ${\textstyle 0<p<1}$ and the sets of the ${\textstyle p}$-sumable series (698-1047-1747220434181-341-341-213) in the real numbers.

$$\ell _{p}(\mathbb {R} ):=\left\{(a_{n})_{n\in \mathbb {N} }\ :\ \|a\|_{p}:=\sum _{n=1}^{\infty }|a_{n}|^{p}<\infty \right\}$$

${\textstyle \|\cdot \|_{p}}$ is a ${\textstyle p}$ standard on the ${\textstyle \mathbb {R} }$-vector space ${\textstyle \ell _{p}(\mathbb {R} )}$.

Definition: p-normality

Let${\textstyle (V,{\mathcal {T}})}$ means ${\textstyle p}$-normally or locally limited with the concavity constant ${\textstyle p}$ if a ${\textstyle p}$-standard

$$\left\|\cdot \right\|:V\longrightarrow \mathbb {K}$$,

exists which produces the topology on ${\textstyle V}$ (formal ${\textstyle (V,\left\|\cdot \right\|)\in {\mathcal {P}}}$).

Definition: p-seminorm

Let${\textstyle V}$ a topological vector space above the field ${\textstyle \mathbb {K} }$ and ${\textstyle 0<p\leq 1}$. A functional ${\textstyle \left\|\cdot \right\|:V\longrightarrow \mathbb {K} }$ is called ${\textstyle p}$-seminorm on ${\textstyle V}$ with (698-1047220434181-341-230) as a concavity constant, if (698-1047-1747220434181-3419) meets the following conditions: (PH1) ${\textstyle \displaystyle \forall _{\displaystyle x\in V}:\left\|x\right\|\geq 0}$ (PH2) ${\textstyle \displaystyle \forall _{\displaystyle x\in V,\lambda \in \mathbb {K} }:\left\|\lambda \cdot x\right\|=|\lambda |^{p}\cdot \left\|x\right\|}$ (PH3) ${\textstyle \displaystyle \forall _{\displaystyle x,y\in V}:\left\|x+y\right\|\leq \left\|x\right\|+\left\|y\right\|}$

Note: p-seminorm

If (PN2) is not valid in the definition of the ${\textstyle p}$ standard, ${\textstyle \left\|\cdot \right\|}$ (698-1047-1747220434181-341-341-237)-seminorm with (698-1047-1747220434181-341-341-238) is considered a concavity constant. Analogously to the standard semi, a single ${\textstyle p}$-seminorm cannot separate the points in the topological vector space (house village property T2).

Multiplikativ pseudokonvex - Submultiplicivity of p-seminorm

A ${\textstyle p}$ seminorm is submultiplicative with a stiffness constant ${\textstyle C>0}$, if applicable for all ${\textstyle v_{1},v_{2}\in V}$:

$$\left\|v_{1}\cdot v_{2}\right\|\leq C\cdot \left\|v_{1}\right\|\cdot \left\|v_{2}\right\|$$

${\textstyle C}$ are called the stiffness constant of the multiplication. The ${\textstyle p}$ seminorm ${\textstyle \left\|\cdot \right\|}$ can be replaced by an equivalent ${\textstyle p}$ seminorm ${\textstyle \left\|\cdot \right\|_{1}}$, for which ${\textstyle C=1}$ is replaced.

Definition: pseudokonvexer vector space

Let${\textstyle (V,{\mathcal {T}})}$ means pseudoconvex if the topology ${\textstyle {\mathcal {T}}}$ is produced by a system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ of ${\textstyle p}$ seminorms which has the following properties.

$$\left\|\cdot \right\|_{\alpha }:V\longrightarrow \mathbb {K} {\mbox{ ist }}p_{\alpha }{\mbox{-homogen mit }}\alpha \in {\mathcal {A}}{\mbox{ und }}p_{\alpha }\in (0,1]$$,

Formally listed ${\textstyle (V,\left\|\cdot \right\|_{\mathcal {A}})\in {\mathcal {PC}}}$.

Note: topology-producing p-standard

A ${\textstyle p}$ standard is topology-producing for topology ${\textstyle {\mathcal {T}}\subseteq \wp (V)}$ if the following condition applies:

$${\forall }_{U\in {\mathcal {T}}}\forall _{v_{o}\in U}\exists _{\varepsilon >0}:\,B_{\varepsilon }^{\left\|\cdot \right\|}(v_{o}):=\left\{v\in V\,:\,\left\|x-x_{o}\right\|<\varepsilon \right\}$$

The ${\textstyle \varepsilon }$ balls are further used for the characterization of the stiffness.

Definition: Epsilon balls of p-gauge functionals

Let${\textstyle V}$ a vector space and ${\textstyle \left\|\cdot \right\|_{\alpha }}$ a (698-1047220434181-341-259)-gauge functional to (698-1047220434181-341-260), then the ${\textstyle \varepsilon }$-261)-ball is defined by (69847

$${\mathrm {B} }_{\varepsilon }^{\alpha }(v):=\{z\in V\,:\,\left\|z-v\right\|_{\alpha }<\varepsilon \}$$

Definition: Quasinorm

Let${\textstyle V}$ a topological vector space above the field ${\textstyle \mathbb {K} }$. A functional

$$\left\|\cdot \right\|:V\longrightarrow \mathbb {K}$$

is called quasinorm on ${\textstyle V}$, if ${\textstyle \left\|\cdot \right\|}$ meets the following conditions:

• (QN1) ${\textstyle \displaystyle \forall _{\displaystyle x\in V}:\left\|x\right\|\geq 0}$
• (QN2) ${\textstyle \displaystyle \left\|x\right\|=0\Longrightarrow x=0}$
• (QN3) ${\textstyle \displaystyle \forall _{\displaystyle x\in V,\lambda \in \mathbb {K} }:\left\|\lambda \cdot x\right\|=|\lambda |\cdot \left\|x\right\|}$
• (QN4) ${\textstyle \displaystyle \exists _{\displaystyle K>0}\forall _{\displaystyle x,y\in V}:\left\|x+y\right\|\leq K\cdot \left(\left\|x\right\|+\left\|y\right\|\right)}$

Definition: semi-norm

A function ${\textstyle \left\|\cdot \right\|:V\longrightarrow \mathbb {K} }$ on a vector space ${\textstyle V}$ above the field ${\textstyle \mathbb {K} }$ means quasi-shalf standard with constant stiffness of the addition ${\textstyle K\geq 1}$ if ${\textstyle \left\|\cdot \right\|}$ meets the following conditions:

• (QH1) ${\textstyle \displaystyle \forall _{\displaystyle x\in V}:\left\|x\right\|\geq 0}$
• (QH2) ${\textstyle \displaystyle \forall _{\displaystyle x\in V,\lambda \in \mathbb {K} }:\left\|\lambda \cdot x\right\|=|\lambda |\cdot \left\|x\right\|}$
• (QH3) ${\textstyle \displaystyle \exists _{\displaystyle K>0}\forall _{\displaystyle x,y\in V}:\left\|x+y\right\|\leq K\cdot \left(\left\|x\right\|+\left\|y\right\|\right)}$

Note: Quasinorm - semi-norm

Analogue to standard semis and norms or ${\textstyle p}$-norms and ${\textstyle p}$-seminorms a quasi-semi-norm (698-1047-1747220434181-34181-341-284) with a continuity constant (698-1047-1747220434181-34285) of the addition, if no longer applies (Q.

Note: Stability constant

The stiffness constant is related to the Konkavitätskonstante einer (698-1047-1747220434181-341-286)-Norm bzw. (698-1047-1747220434181-341-287)-seminorm. This shows the Korrespondenzlemma für (698-1047-1747220434181-341-288)-seminorms

Convergence over nets

Let(698-10471747220434181-341-289) has a topologischer space, ${\textstyle x_{0}\in X}$ and (698-1047220434181-341-291) has a net in ${\textstyle X}$ with an index set (698-10471747220434181-341-293). Convergence over nets is defined as follows:

$${\stackrel {\mathcal {T}}{\lim _{i\to \infty }}}x_{i}\,:\Longleftrightarrow \,(x_{i})_{i\in I}{\stackrel {\mathcal {T}}{\longrightarrow }}x_{0}\,:\Longleftrightarrow \,\forall _{U\in {\mathfrak {U}}_{\mathcal {T}}(x_{0})}\exists _{i_{U}\in I}\forall _{i\geq i_{U}}:\,x_{i}\in U$$

Definition: Algebre classes

The distinction according to algebras classes is essential for the investigation of permanently singular elements, since the invertability in an algebra extension depends on the class ${\textstyle {\mathcal {K}}}$.

Notation 1: Algebras classes

Let${\textstyle {\mathcal {K}}}$ a class of topological algebras and ${\textstyle \mathbb {K} }$ a field, then subclasses of topological algebras are denoted by the following symbols:

• ${\textstyle {\mathcal {K}}_{e}}$ Class of unital algebras in ${\textstyle {\mathcal {K}}}$;
• ${\textstyle {\mathcal {K}}^{k}}$ Class of commutative algebras in ${\textstyle {\mathcal {K}}}$, commutative refers to the multiplication in the algebras.
• ${\textstyle {\mathcal {K}}(\mathbb {K} )}$ Class of topological algebras over ${\textstyle \mathbb {K} }$ in ${\textstyle {\mathcal {K}}}$;

Notation 2: Algebras classes

• ${\textstyle {\mathcal {T}}}$ Class of all topological algebras;
• ${\textstyle {\mathcal {B}}}$ Class of all Banach algebras (full, normed);
• ${\textstyle {\mathcal {LC}}}$ class of local convex algebras; i.e. topology generated by a system of seminorms;
• ${\textstyle {\mathcal {MLC}}}$ Class of multiplicative local convex algebras;

Notation 3: Algebras classes

• ${\textstyle {\mathcal {P}}}$ Class of the ${\textstyle p}$ standardizable algebras or locally limited algebras;
• ${\textstyle {\mathcal {PC}}}$ Class of pseudoconvex algebras; & i.e. topology produced by a system of ${\textstyle p}$-seminorms;
• ${\textstyle {\mathcal {MPC}}}$ Class of the multiplicative pseudoconvexen algebras.

Note: Pseudokonvexe rooms

For pseudoconvexe algebras, the ${\textstyle {\mathcal {PC}}}$ system can also be used from the appropriate semi-standards. With the Korrespondenzsatz für (698-1047-1747220434181-341-314)-seminorms the relationship of ${\textstyle p}$-seminorms and quasi-seminorms. Other not all ${\textstyle p}$-seminorms ${\textstyle \left\|\cdot \right\|_{\alpha }}$ have the same Concavity constant (see definition gauge functional) ${\textstyle 0<p_{\alpha }\leq 1}$, i.e. ${\textstyle p_{\alpha }}$

$$|\lambda |^{p_{\alpha }}\cdot \left\|x\right\|_{\alpha }=\left\|\lambda \cdot x\right\|_{\alpha }{\mbox{ für alle }}x\in A{\mbox{ und }}\lambda \in \mathbb {K} .$$

Task 1: Norm

Draw the ${\textstyle \varepsilon }$ ball in ${\textstyle V:=\mathbb {R} ^{2}}$ with (698-1047220434181-341-322) and

$${\begin{array}{rrcl}\left\|\cdot \right\|_{p}:&\mathbb {R} ^{2}&\rightarrow &\mathbb {R} _{o}^{+}\\&x&\mapsto &\left\|\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)\right\|_{p}:={\sqrt[{p}]{|x_{1}|^{p}+|x_{2}|^{p}}}\end{array}}$$

drawing the edge of the ${\textstyle \varepsilon }$ balls ${\textstyle B_{\varepsilon }^{\left\|\cdot \right\|_{p}}\left(x\right)}$ with respect to the standard ${\textstyle \left\|\cdot \right\|_{p}}$ with

• ${\textstyle p=1}$ ${\textstyle \varepsilon =1}$ and ${\textstyle x=\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)=\left({\begin{array}{c}0\\0\end{array}}\right)}$
• ${\textstyle p=3}$ ${\textstyle \varepsilon =1}$ and ${\textstyle x=\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)=\left({\begin{array}{c}0\\0\end{array}}\right)}$

Task 2: p-standard

Draw the ${\textstyle \varepsilon }$ ball in ${\textstyle V:=\mathbb {R} ^{2}}$ with ${\textstyle p\leq 1}$ and

$${\begin{array}{rrcl}\left\|\cdot \right\|_{p}:&\mathbb {R} ^{2}&\rightarrow &\mathbb {R} _{o}^{+}\\&x&\mapsto &\left\|\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)\right\|_{p}:=|x_{1}|^{p}+|x_{2}|^{p}\end{array}}$$

drawing the edge of the ${\textstyle \varepsilon }$ balls ${\textstyle B_{\varepsilon }^{\left\|\cdot \right\|_{p}}\left(x\right)}$ with respect to the standard ${\textstyle \left\|\cdot \right\|_{p}}$ with

• ${\textstyle p={\frac {1}{4}}}$ (698-1047-1747220434181-341-341-339) and ${\textstyle x=\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)=\left({\begin{array}{c}0\\0\end{array}}\right)}$
• ${\textstyle p={\frac {1}{2}}}$ (698-1047-1747220434181-341-341-342) and ${\textstyle x=\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)=\left({\begin{array}{c}0\\0\end{array}}\right)}$

See also

• nets
• norms
• Korrespondenzlemma für (698-1047-1747220434181-341-344)-seminorms
• Theorem of continuity für lineare mappings
• sequence of continuity
• MLC-Regularität
• MPC-Regularität
• LC-Regularität
• PC-Regularität

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Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Kurs:Topologische Invertierbarkeitskriterien/Gaugefunktionale
• URL: https://de.wikiversity.org/wiki/Kurs:Topologische%20Invertierbarkeitskriterien/Gaugefunktionale
• Date: 5/14/2025 13:18