Inverse-producing extensions of Topological Algebras/Gauge functionals

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

$${\displaystyle \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 }

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 ${\displaystyle v_{o}\in V}$:

$${\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 }

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.

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

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, ...)

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.

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

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

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

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 ${\textstyle p=1}$.

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

The vector space ${\textstyle V_{p}}$ and the ${\textstyle p}$-gauge functional ${\displaystyle \|x\|_{p}:=\sum _{n=1}^{\infty }|x_{n}|^{p}}$ 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 ${\textstyle p,q\in (0,1]}$?

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^[1] that induces the same open sets on the corresponding ${\displaystyle p}$-seminorm. The triangle inequation for the ${\displaystyle p}$-homogeneous ${\displaystyle p}$-seminorm has a quasi-seminorm with a corresponding concavity constant ${\displaystyle K}$ with:

${\displaystyle \|x+y\|\leq K\cdot (\|x\|+\|y\|)}$

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.

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

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

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 ${\displaystyle p}$-gauge functionals on ${\displaystyle V}$. 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) ${\textstyle \forall _{\beta \in I_{2}}\exists _{\alpha \in I_{1},C_{\alpha }>0}\forall _{v\in V}:\,\,\|v\|_{\beta }\leq C_{\alpha }\cdot \|v\|_{\alpha }}$

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:

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

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

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

Let ${\textstyle (V,{\mathcal {T}})}$ a topological vector space with the system of open sets ${\textstyle {\mathcal {T}}}$ on ${\displaystyle V}$. 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) ${\textstyle {\forall }_{U\in {\mathcal {T}}}\forall _{x_{o}\in U}\exists _{\alpha \in I,\varepsilon >0}:\,B_{\varepsilon }^{\alpha }(x_{o}):=\left\{x\in V\,:\,\left\|x-x_{o}\right\|_{\alpha }<\varepsilon \right\}\subseteq U}$

• (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.

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

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

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

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

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

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).

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 ${\textstyle U_{\alpha }}$.

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

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

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

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 ${\textstyle v_{1}-v_{2}\not =0_{V}}$.

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

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

${\textstyle C}$ 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.

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

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

and the submultiplicity is obtained via:

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

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

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

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.

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:

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

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.

$${\displaystyle \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 ${\textstyle \ell _{p}(\mathbb {R} )}$.

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

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

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

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

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).

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

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

${\textstyle C}$ 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.

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.

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

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

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

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

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

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

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

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

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

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) ${\textstyle \displaystyle \exists _{\displaystyle K>0}\forall _{\displaystyle x,y\in V}:\left\|x+y\right\|\leq K\cdot \left(\left\|x\right\|+\left\|y\right\|\right)}$

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) ${\textstyle \displaystyle \exists _{\displaystyle K>0}\forall _{\displaystyle x,y\in V}:\left\|x+y\right\|\leq K\cdot \left(\left\|x\right\|+\left\|y\right\|\right)}$

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.

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

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 ${\textstyle {\mathcal {K}}}$;

• ${\textstyle {\mathcal {T}}}$ 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;

• ${\textstyle {\mathcal {P}}}$ 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.

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

1. ↑ Kalton, N. (1986). "Plurisubharmonic functions on quasi-Banach spaces". Studia Mathematica (Institute of Mathematics, Polish Academy of Sciences) 84 (3): 297–324. doi:10.4064/sm-84-3-297-324. ISSN 0039-3223. https://kaltonmemorial.missouri.edu/assets/docs/sm1986b.pdf.