Inverse producing extension/Topologization Lemma of Algebras

The topologization lemma for algebras allows the description of the stiffness of the Verknüpfungen in a topologischen Algebra over gauge functional. You can represent this learning resource as Wiki2Reveal slides.

Sei ${\textstyle A}$ an algebra. ${\textstyle A}$ is precisely a topological algebra ${\textstyle (A,{\mathcal {T}})}$, which meets the Hausdorff property if the Topologie ${\textstyle {\mathcal {T}}}$ can be produced by a measuring functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ with the properties (A1)-(A5).

• (A1) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}},x\in A}:\left\|x\right\|_{\alpha }\geq 0}$
• (A2) ${\textstyle \displaystyle \left(\forall _{\alpha \in {\mathcal {A}}}:\left\|x\right\|_{\alpha }=0\right)\Longrightarrow x=0_{A}}$
• (A3) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}},x\in A,\lambda \in \mathbb {K} }:\left\|\lambda \cdot x\right\|_{\alpha }=|\lambda |\cdot \left\|x\right\|_{\alpha }}$
• (A4) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}}}\exists _{\beta \in {\mathcal {A}}}\forall _{x,y\in A}:\left\|x+y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }+\left\|y\right\|_{\beta }}$
• (A5) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}}}\exists _{\beta \in {\mathcal {A}}}\forall _{x,y\in A}:\left\|x\cdot y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }\cdot \left\|y\right\|_{\beta }}$

Sei ${\textstyle V}$ a vector space. ${\textstyle V}$ is precisely then a topological vector space ${\textstyle (V,{\mathcal {T}})}$, which meets the Hausdorff property if the topology ${\textstyle {\mathcal {T}}}$ can be produced by a measuring functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ having the properties (V1)-(V4).

• (V1) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}},x\in V}:\left\|x\right\|_{\alpha }\geq 0}$
• (V2) ${\textstyle \displaystyle \left(\forall _{\alpha \in {\mathcal {A}}}:\left\|x\right\|_{\alpha }=0\right)\Longrightarrow x=0_{V}}$
• (V3) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}},x\in V,\lambda \in \mathbb {K} }:\left\|\lambda \cdot x\right\|_{\alpha }=|\lambda |\cdot \left\|x\right\|_{\alpha }}$
• (V4) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}}}\exists _{\beta \in {\mathcal {A}}}\forall _{x,y\in V}:\left\|x+y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }+\left\|y\right\|_{\beta }}$

The topologischen Vektorraumes ${\textstyle (V,{\mathcal {T}})}$ are missing the stetig inner probation of the multiplication ${\textstyle \cdot :V\times V\to V}$. Therefore, for topologischen Vektorraumes, only four characterizing properties (V1)-(V4) are obtained. The evidence of the properties is analogous to the proof of the topologization Lemma for topologische Algebren.

If the topological algebra does not meet the Hausdorff property, (A2) does not apply and two points ${\textstyle x_{1}\not =x_{2}}$ cannot be separated by the functional measuring system. Unless otherwise indicated, the topologische Algebren are house village f'sch.

For proof, you have to show two directions,

• that for a topologischen Algebra ${\textstyle (A,{\mathcal {T}})}$ a system of measuring functionals ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ exists with the following properties (A1)-(A5) and
• Conversely, a system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ of measuring functionals with (A1)-(A5) produces a Topologie ${\textstyle {\mathcal {T}}}$ on ${\textstyle A}$, which makes ${\textstyle (A,{\mathcal {T}})}$ a topologischen Algebra.

Without limitation, we use a measuring functional system that is completely extended with respect to the radius of the ${\textstyle \varepsilon }$ balls. This means that the measuring functional system is also extended by equivalent measuring functional of the form ${\textstyle \|\cdot \|_{(\alpha ,\varepsilon )}:=\varepsilon \cdot \|\cdot \|_{\alpha }}$. This saves the use of constants in the following proof.

The evidence suggests that in linear or bilinear Abbildung, the stiffness has to be detected only at one point (here zero vector) in order to demonstrate the stiffness in the whole definition range of the (bi-)linear Abbildung (see also Stetigkeitssatz für lineare Abbildungen))

The following evidence establishes the following relationship with Topologie and Mengen for the properties (A1)-(A5).

• (A1) ${\textstyle \left\|x\right\|_{\alpha }\geq 0}$ - Definition of the Minkowski function for absorbent Mengen
• (A2) ${\textstyle \displaystyle \left(\forall _{\displaystyle \alpha \in {\mathcal {A}}}:\left\|x\right\|_{\alpha }=0\right)\Longrightarrow x=0_{A}}$ Household Village Fact of the Topologie
• (A3) ${\textstyle \left\|\lambda \cdot x\right\|_{\alpha }=|\lambda |\cdot \left\|x\right\|_{\alpha }}$ Stability of Multiplikation mit Skalaren
• (A4) ${\textstyle \left\|x+y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }+\left\|y\right\|_{\beta }}$ Stability of the addition
• (A5) ${\textstyle \left\|x\cdot y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }\cdot \left\|y\right\|_{\beta }}$ Strength of multiplication

Topological Vektorräumen own a neighborhood system of the zero vector ${\textstyle \{U_{\alpha }\}_{\alpha \in {\mathcal {A}}}}$ circular Mengen. It is then considered as a system of the Minkowski functional ${\textstyle \|\cdot \|_{\alpha }}$ with ${\textstyle \alpha \in {\mathcal {A}}}$ of this circular Zero environments ${\textstyle \{U_{\alpha }\}_{\alpha \in {\mathcal {A}}}}$. Only two directions of evidence shall be shown:

• A topologischen Algebra ${\textstyle (A{\mathcal {T}})}$ is given and the properties (A1)-(A5) are to be shown and vice versa
• for given Topologie-producing Gaugeunktional with the properties ${\textstyle (A1)-(A5)}$ for ${\textstyle {\mathcal {T}}}$, the algebra becomes a topologischen Algebra.

A topologischen Algebra ${\textstyle (A,{\mathcal {T}})}$ is given and the properties (A1) are shown.

With the circular zero environment ${\textstyle U_{\alpha }\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$, the Minkowski function is defined:

$${\displaystyle \|x\|_{\alpha }:=\inf\{\lambda >0\,:\,x\in \lambda \cdot U_{\alpha }\}}$$.

The property of non-negativity follows from the definition of Minkowski functionals, since an infimum is formed via positive numbers, with which an absorbent (circular) Menge ${\textstyle U_{\alpha }\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$ still captures an element ${\textstyle x\in A}$ ((698-1047-17615836065)). An infimum of positive numbers is at least not negative (i.e. ${\textstyle 0\leq \|x\|_{\alpha }\leq \lambda }$.

Conversely, if a Topologie-producing measuring functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is given on an algebra that properties ${\textstyle (A1)-(A5)}$, then there is a neighborhood system of the zero vector ${\textstyle \{U_{\alpha }\}_{\alpha \in {\mathcal {A}}}}$ made of circular Mengen.

The neighborhood system of the zero vector ${\textstyle \{\varepsilon \cdot U_{\alpha }\}_{\alpha \in {{\mathcal {A}},\lambda >0}}}$ is defined with:

$${\displaystyle U_{\alpha }:=B_{1}^{\alpha }(0_{A}):=\left\{x\in A\,:\,\left\|x\right\|_{\alpha }<1\right\}}$$

The Menge ${\textstyle U_{(\alpha ,\varepsilon )}:=\varepsilon \cdot U_{\alpha }\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$ is a zero environment because ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is the Topologie for ${\textstyle {\mathcal {T}}}$.

It is possible to confine itself here to the 1-Umgebungen because the measurement functional system with respect to the ${\textstyle \varepsilon }$ wheel is completed without restriction, i.e. it may have been by equivalent measuring functionale of the form ${\textstyle \|\cdot \|_{(\alpha ,\varepsilon )}:=\varepsilon \cdot \|\cdot \|_{\alpha }}$ if applicable.

Furthermore, the Menge ${\textstyle U_{\alpha }\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$ kreisförmig is applicable because with ${\textstyle x\in U_{\alpha }}$, ${\textstyle \lambda \in \mathbb {K} }$ and ${\textstyle |\lambda |\leq 1}$ with (A3):

$${\displaystyle \left\|\lambda \cdot x\right\|_{\alpha }{\stackrel {(A3)}{=}}|\lambda |\cdot \left\|x\right\|_{\alpha }\leq \left\|x\right\|_{\alpha }<1}$$

Thus, ${\textstyle \left\|\lambda \cdot x\right\|_{\alpha }<1}$ also applies ${\textstyle \lambda \cdot x\in U_{\alpha }}$ and ${\textstyle U_{\alpha }}$ is kreisförmig.

For any vector ${\textstyle x_{o}\in A}$, the neighborhood base ${\textstyle \{B_{\varepsilon }^{\alpha }(x_{o})\}_{\alpha \in {{\mathcal {A}},\varepsilon >0,x_{o}\in A}}}$ is defined as follows:

$${\displaystyle B_{\varepsilon }^{\alpha }(x_{o}):=\left\{x\in A\,:\,\left\|x-x_{o}\right\|_{\alpha }<\varepsilon \right\}}$$

The Menge ${\textstyle B_{\varepsilon }^{\alpha }(x_{o}):=U_{\alpha }\in {\mathfrak {U}}_{\mathcal {T}}(x)}$ is a neighborhood of ${\textstyle x_{o}}$, because ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is the Topologie for ${\textstyle {\mathcal {T}}}$.

${\textstyle {\mathcal {T}}}$ is now the smallest Topologie produced by the Mengenystem ${\textstyle \{B_{\varepsilon }^{\alpha }(x_{o})\}_{\alpha \in {\mathcal {A}},\varepsilon >0,x_{o}\in A}}$. In the further course it is now necessary to prove that the Multiplikation mit Skalaren, addition and multiplication are stetig on ${\textstyle (A,{\mathcal {T}})}$ when the measuring system is functional.

A topologischen Algebra ${\textstyle (A,{\mathcal {T}})}$ is given and the property (A2) is to be shown.

The topological algebra ${\textstyle (A,{\mathcal {T}})}$ according to definition Hausdorff space. It is considered ${\textstyle x\in A}$ with ${\textstyle x\not =0_{A}}$. There is then a neighborhood ${\textstyle U_{0}\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$ and ${\textstyle U_{1}\in {\mathfrak {U}}_{\mathcal {T}}(x)}$ with ${\textstyle U_{0}\cap U_{1}=\emptyset }$.

Since in a topologischer Vektorraum (Algebra) a neighborhood system of the zero vector from circular Mengen ${\textstyle \{U_{\alpha }\}_{\alpha \in {\mathcal {A}}}}$ exists, there is a${\textstyle \alpha \in {\mathcal {A}}}$ with ${\textstyle U_{\alpha }\subseteq U_{0}}$. Since ${\textstyle U_{0}\cap U_{1}=\emptyset }$ and ${\textstyle x\in U_{1}}$ apply, ${\textstyle x\notin U_{0}}$ follows and thus the definition of Minkowski functionals

$${\displaystyle \|x\|_{\alpha }:=\inf\{\lambda >0\,:\,x\in \lambda \cdot U_{\alpha }\}\geq 1}$$ and there is a ${\textstyle \alpha \in {\mathcal {A}}}$ with ${\textstyle \|x\|_{\alpha }\not =0}$

Conversely, if on an algebra a Topologie-producing measurement functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is given that properties ${\textstyle (A1)-(A5)}$ are fulfilled, then the House Village F property'' for the Topologie produced (698-1047-1761583604265-1241.

Sei ${\textstyle x_{1},x_{2}\in A}$ with ${\textstyle x_{1}\not =x_{2}}$ be selected as desired. Now the Topologie-producing measuring functionals ${\textstyle \|\cdot \|_{\alpha }}$ with ${\textstyle \alpha \in {\mathcal {A}}}$ are used to produce two neighborhoods ${\textstyle U_{1}\in {\mathfrak {U}}_{\mathcal {T}}(x_{1})}$ and ${\textstyle U_{2}\in {\mathfrak {U}}_{\mathcal {T}}(x_{2})}$, for which Failed to parse (unknown function "\emptyMenge"): {\textstyle U_1 \cap U_2 = \emptyMenge } applies.

The Topologie of the algebra is now produced by a measurement functional system having the property (A1)-(A5). Condition (A2) is now applied to ${\textstyle x:=x_{1}-x_{2}}$:

$${\displaystyle \displaystyle \left(\forall _{\displaystyle \alpha \in {\mathcal {A}}}:\left\|x\right\|_{\alpha }=0\right)\Longrightarrow x=0_{A}}$$

The double posting of (A2) for ${\textstyle x=x_{1}-x_{2}}$ gives a ${\textstyle \alpha \in {\mathcal {A}}}$ with ${\textstyle \left\|x_{1}-x_{2}\right\|_{\alpha }>0}$, since:

$${\displaystyle x\not =0_{A}\Longrightarrow \displaystyle \left(\exists _{\displaystyle \alpha \in {\mathcal {A}}}:\left\|x\right\|_{\alpha }>0\right)}$$

We now apply the above double position to ${\textstyle x:=x_{1}-x_{2}\not =0}$. Therefore there is a ${\textstyle \alpha \in {\mathcal {A}}}$ with

$${\displaystyle 0<\lambda :=\|x\|_{\alpha }=\|x_{1}-x_{2}\|_{\alpha }=|-1|\cdot \|x_{1}-x_{2}\|_{\alpha }=\|x_{2}-x_{1}\|_{\alpha }}$$

With the condition (A4) there is a ${\textstyle \alpha \in {\mathcal {A}}}$ ${\textstyle \beta \in {\mathcal {A}}}$ with

$${\displaystyle \displaystyle \forall _{\displaystyle x,y\in A}:\left\|x+y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }+\left\|y\right\|_{\beta }}$$

Now use ${\textstyle r:={\frac {\lambda }{3}}}$ to Menge the neighborhoods ${\textstyle U_{1}:=B_{r}^{\beta }(x_{1})}$ and ${\textstyle U_{2}:=B_{r}^{\beta }(x_{2})}$. Because of ${\textstyle \|x_{1}-x_{1}\|_{\beta }=0\leq r}$ and ${\textstyle \|x_{2}-x_{2}\|_{\beta }=0\leq r}$, in particular ${\textstyle x_{1}\in U_{1}}$ and ${\textstyle x_{2}\in U_{2}}$. In order to prove the house village f property, it is now possible to show Failed to parse (unknown function "\emptyMenge"): {\textstyle U_1 \cap U_2 = \emptyMenge } .

It is now assumed that the cut ${\textstyle U_{1}\cap U_{2}}$ is not the empty Menge and thus a ${\textstyle x_{o}\in U_{1}\cap U_{2}}$ exists, for which:

$${\displaystyle {\begin{array}{rcl}0&<&\lambda :=\|x\|_{\alpha }=\|x_{1}-x_{2}\|_{\alpha }=\|(x_{1}-x_{0})+(x_{0}-x_{2})\|_{\alpha }\\&\leq &\|x_{1}-x_{0}\|_{\beta }+\|x_{0}-x_{2}\|_{\beta }\leq {\frac {\lambda }{3}}+{\frac {\lambda }{3}}={\frac {2\cdot \lambda }{3}}\end{array}}}$$

Objection, for ${\textstyle 0<\lambda <{\frac {2}{3}}\cdot \lambda }$. So applies Failed to parse (unknown function "\emptyMenge"): {\textstyle U_1 \cap U_2 = \emptyMenge } and the Topologie of the algebra is Household village.

The condition (A2) therefore supplies the algebra Hausdorff property ${\textstyle A}$ on the basis of the condition (A4). If the Hausdorff property is not required, the topologization of Lemma eliminates the property (A2).

A topologischen Algebra ${\textstyle (A,{\mathcal {T}})}$ is given and the properties (A3) are shown.

In a topological vectoraum ${\textstyle (A,{\mathcal {T}})}$ (and of course also in a topologischen Algebra) there is a neighborhood system of the zero vector from circular Mengen. Considering the corresponding Minkowski functional, these are homogeneous in circular Mengen (i.e. measure functional). For any ${\textstyle \varepsilon >0}$, Failed to parse (unknown function "\Mengeminus"): {\textstyle \lambda \in \mathbb{K}\Mengeminus \{0\} } and the circularity of ${\textstyle U_{\alpha }}$:

$${\displaystyle x\in (\|x\|_{\alpha }+\varepsilon )\cdot U_{\alpha }\,\Rightarrow \,{\frac {|\lambda |}{\lambda }}\cdot x\in (\|x\|_{\alpha }+\varepsilon )\cdot U_{\alpha }}$$

This applies to ${\textstyle \left|{\frac {|\lambda |}{\lambda }}\right|=1}$

It also applies to ${\textstyle \lambda \cdot x}$, ${\textstyle \lambda \not =0}$ and ${\textstyle \lambda \cdot x\in (\|\lambda \cdot x\|_{\alpha }+\varepsilon )\cdot U_{\alpha }}$

$${\displaystyle \,|\lambda |\cdot x={\frac {|\lambda |}{\lambda }}\cdot \lambda \cdot x\in (\|\lambda \cdot x\|_{\alpha }+\varepsilon )\cdot U_{\alpha }}$$

All in all ${\textstyle x\in \left({\frac {\|\lambda \cdot x\|_{\alpha }+\varepsilon }{|\lambda |}}\right)\cdot U_{\alpha }}$ and about infimum formation are obtained. ${\textstyle \varepsilon >0}$ is obtained: ${\textstyle \|x\|_{\alpha }\leq {\frac {\|\lambda \cdot x\|_{\alpha }}{|\lambda |}}}$ or ${\textstyle |\lambda |\cdot \|x\|_{\alpha }\leq \|\lambda \cdot x\|_{\alpha }}$.

The reverse equation ${\textstyle |\lambda |\cdot \|x\|_{\alpha }\geq \|\lambda \cdot x\|_{\alpha }}$ is obtained with the circularity of ${\textstyle U_{\alpha }}$ and ${\textstyle \lambda \cdot x\in (\|\lambda \cdot x\|_{\alpha }+\varepsilon )\cdot U_{\alpha }}$:

$${\displaystyle |\lambda |\cdot \underbrace {{\frac {\lambda }{|\lambda |}}\cdot x} _{\in (\|x\|_{\alpha }+\varepsilon )\cdot U_{\alpha }}\in |\lambda |\cdot (\|x\|_{\alpha }+\varepsilon )\cdot U_{\alpha }}$$

Thus, the infimum formation of ${\textstyle \varepsilon >0}$ is also obtained ${\textstyle \|\lambda \cdot x\|_{\alpha }\leq |\lambda |\cdot \|x\|_{\alpha }}$.

Evidence step (A3.2) and (A3.3) together delivers equality ${\textstyle |\lambda |\cdot \|x\|_{\alpha }=\|\lambda \cdot x\|_{\alpha }}$.

Conversely, if a Topologie-producing measurement functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is given on an algebra, that properties ${\textstyle (A1)-(A5)}$ are fulfilled, then the stiffness of the Multiplikation mit Skalaren is shown with respect to ${\textstyle {\mathcal {T}}}$.

Conversely, if there is a zero Folge ${\textstyle (\lambda _{n})_{n\in \mathbb {N} }}$ and a Netzes ${\textstyle (x_{i})_{i\in I}}$ which is converged against the zero vector ${\textstyle 0_{A}}$ in the Topologie ${\textstyle 0_{A}}$, the following applies:

$${\displaystyle \displaystyle \forall _{\varepsilon >0,\alpha \in {\mathcal {A}}}\exists _{n_{\varepsilon }\in \mathbb {N} ,i_{\alpha }\in I}\forall _{n\geq n_{\varepsilon },i\geq i_{\alpha }}\,:\,\,|\lambda _{n}|<\varepsilon \,\wedge x_{i}\in U_{\alpha }}$$

This results in the use of condition (A3):

$${\displaystyle \displaystyle \forall _{\varepsilon >0,\alpha \in {\mathcal {A}}}\exists _{n_{\varepsilon }\in \mathbb {N} ,i_{\alpha }\in I}\forall _{n\geq n_{\varepsilon },i\geq i_{\alpha }}\,:\,\,\|\lambda _{n}\cdot x_{i}\|_{\alpha }{\stackrel {(A3)}{=}}|\lambda _{n}|\cdot \underbrace {\|x_{i}\|_{\alpha }} _{\leq 1}<\varepsilon \cdot 1=\varepsilon }$$

In total ${\textstyle (\lambda _{n}\cdot x_{i})_{(n,i)\in \mathbb {N} \times I}}$ also converged against the zero vector ${\textstyle 0_{A}}$, since for all ${\textstyle U\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$ a circular zero environment Failed to parse (unknown function "\subMengeeq"): {\textstyle U_\alpha \subMengeeq U } exists. For this circular zero environment there is an index barrier ${\textstyle i_{\alpha }\in I}$, from which all Failed to parse (unknown function "\subMengeeq"): {\textstyle x_i \in U_\alpha \subMengeeq U } are located with ${\textstyle i\geq i_{\alpha }}$. For a circular neighborhood system of the zero vector from ${\textstyle U_{\alpha }:=B_{1}^{\alpha }(0_{A})}$ and ${\textstyle |\lambda _{n}|<1}$ is also located in Failed to parse (unknown function "\subMenge"): {\textstyle \lambda_n \cdot x_i \in U_\alpha \subMenge U } (i.e. ${\textstyle \lambda _{n}\cdot x_{i}\in U}$ with ${\textstyle n>n_{\varepsilon }}$. The index barrier exists due to convergence of ${\textstyle (\lambda _{n})_{n\in \mathbb {N} }}$ giving ${\textstyle 0\in \mathbb {K} }$. Overall, this shows the stiffness of the Multiplikation mit Skalaren on ${\textstyle A}$.

A topologischen Algebra ${\textstyle (A,{\mathcal {T}})}$ is given and the properties (A4) are shown.

The stiffness of the addition provides:

Failed to parse (unknown function "\subMenge"): {\displaystyle \forall_{\displaystyle U_{\alpha}\in \mathfrak{U} (0_A)} \exists_{\displaystyle U_\beta\in \mathfrak{U} (0_A)} : U_\beta + U_\beta \subMenge U_\alpha }

In particular, Failed to parse (unknown function "\subMenge"): {\textstyle U_\beta = U_\beta + \{0_A\} \subMenge U_\beta + U_\beta \subMenge U_\alpha } applies because ${\textstyle 0_{A}\in U_{\beta }}$ applies. So... Failed to parse (unknown function "\subMenge"): {\textstyle U_\beta \subMenge U_\alpha } for the corresponding Minkowski functionals are also obtained ${\textstyle \left\|x\right\|_{\alpha }\leq \left\|x\right\|_{\beta }}$ for all ${\textstyle x\in A}$. This means that an absorbing subMenge ${\textstyle U_{\beta }}$ must be inflated, if necessary, more strongly in comparison with ${\textstyle U_{\alpha }}$ in order to capture the corresponding ${\textstyle x\in A}$.

For ${\textstyle \left\|x\right\|_{\beta }=0}$ and ${\textstyle \left\|y\right\|_{\beta }=0}$ is ${\textstyle x,y\in \lambda \cdot U_{\beta }}$ for all ${\textstyle \lambda >0}$. In particular,

Failed to parse (unknown function "\subMenge"): {\displaystyle x+y \in \lambda \cdot U_\beta +\lambda \cdot U_\beta = \lambda \cdot (U_\beta + U_\beta) \subMenge \lambda \cdot U_\alpha }

for all ${\textstyle \lambda >0}$. This gives the searched (un-) equation directly with

$${\displaystyle \left\|x+y\right\|_{\alpha }=0=\underbrace {\left\|x\right\|_{\beta }} _{=0}+\underbrace {\left\|y\right\|_{\beta }} _{=0}}$$ for all ${\textstyle z\in A}$.

For ${\textstyle \left\|x\right\|_{\beta }\not =0}$ and ${\textstyle \left\|y\right\|_{\beta }=0}$, ${\textstyle y\in \lambda \cdot U_{\beta }}$ applies to all ${\textstyle \lambda >0}$. It also applies to all ${\textstyle \varepsilon >0}$ because of infimum definition of the Minkowski function ${\textstyle x\in (\varepsilon +\|x\|_{\beta })\cdot U_{\beta }}$.

The desired inequality is obtained with ${\textstyle y\in \lambda \cdot U_{\beta }}$ for all ${\textstyle \lambda >0}$

Failed to parse (unknown function "\subMengeeq"): {\displaystyle \begin{array}{rcl} x+y & \in &(\varepsilon + \|x\|_\beta )\cdot U_\beta + (\underbrace{\varepsilon + \|x\|_\beta )}_{\lambda :=}\cdot U_\beta \\ & = & (\varepsilon + \|x\|_\beta )\cdot (U_\beta + U_\beta) \\ & \subMengeeq & (\varepsilon + \|x\|_\beta )\cdot U_\alpha \end{array} }

for all ${\textstyle \varepsilon >0}$. So also applies ${\textstyle \left\|x+y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }=\left\|x\right\|_{\beta }+\underbrace {\left\|y\right\|_{\beta }} _{=0}}$.

The statement is obtained analogously for ${\textstyle \left\|x\right\|_{\beta }=0}$ or ${\textstyle \left\|y\right\|_{\beta }\not =0}$.

Seien now ${\textstyle \left\|x\right\|_{\beta }\not =0}$ and ${\textstyle \left\|y\right\|_{\beta }\not =0}$. According to the definition of the Minkowski functionals for the zero environments ${\textstyle U_{\alpha }}$ and ${\textstyle U_{\beta }}$, for each ${\textstyle \varepsilon >0}$:

Failed to parse (unknown function "\subMenge"): {\displaystyle \begin{array}{rcl} \frac{x}{\left\| x \right\| _\beta+\varepsilon} \in U_\beta \, \wedge \, \frac{y}{\left\| y \right\| _\beta+\varepsilon} \in U_\beta &\Longrightarrow& \frac{x}{\left\| x \right\| _\beta+\left\| y \right\| _\beta}, \frac{y}{\left\| x \right\| _\beta+\left\| y \right\| _\beta} \in U_\beta \\ & \Longrightarrow & \frac{x+y}{\left\| x \right\| _\beta+\left\| y \right\| _\beta} \in U_\beta + U_\beta \subMenge U_\alpha \\ &\Longrightarrow& \left\| \frac{x+y}{\left\| x \right\| _\beta+\left\| y \right\| _\beta} \right\| _\alpha \leq 1 \Longrightarrow \mbox{ Beh. (A4). } \end{array} }

The circularity or Homogeneity (A3) of the Minkowski functionals used from the previous evidence.

Conversely, if on an algebra a Topologie-producing measurement functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is given that properties ${\textstyle (A1)-(A5)}$ are fulfilled, then the stiffness of the addition is shown with respect to ${\textstyle {\mathcal {T}}}$.

If the Topologie ${\textstyle {\mathcal {T}}}$ is generated by ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ and the condition (A4), then applies:

$${\displaystyle \forall _{\displaystyle \alpha \in {\mathcal {A}}}\exists {\beta \in {\mathcal {A}}}\forall _{\displaystyle x,y\in A}:\left\|x+y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }+\left\|y\right\|_{\beta },}$$

Then the addition to ${\textstyle A}$ is stetig, because for each zero environment ${\textstyle U_{\alpha }}$ from the circular neighborhood system of the zero vector is valid with ${\textstyle 0<\varepsilon <{\frac {1}{2}}}$, ${\textstyle x,y\in {\rm {B}}_{\varepsilon }^{\beta }(0)=:U_{\beta }}$ and (698-1047-176236065-341

$${\displaystyle \left\|x+y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }+\left\|y\right\|_{\beta }\leq \varepsilon +\varepsilon <{\frac {1}{2}}+{\frac {1}{2}}=1}$$

In total ${\textstyle x+y\in U_{\alpha }}$ and for any circular zero environment ${\textstyle U_{\alpha }}$ there is ${\textstyle U_{\beta }\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$ with Failed to parse (unknown function "\subMengeeq"): {\textstyle U_\beta + U_\beta \subMengeeq U_\alpha } .

A topologischen Algebra ${\textstyle (A,{\mathcal {T}})}$ is given and the properties (A5) are shown.

The rigidity of the multiplication is obtained as in ${\textstyle (A4)}$

Failed to parse (unknown function "\subMenge"): {\displaystyle \forall_{\displaystyle U_{\alpha}\in \mathfrak{U} (0)} \exists_{\displaystyle U_\beta\in \mathfrak{U} (0)} : U_\beta \cdot U_\beta = U_\beta^{^2} \subMenge U_\alpha }

Sei ${\textstyle \varepsilon >0}$ selected as desired and the claim ${\textstyle (A5)}$, for:

Failed to parse (unknown function "\subMenge"): {\displaystyle \frac{x}{\left\| x \right\| _\beta+\varepsilon}\cdot \frac{y}{\left\| y \right\| _\beta+\varepsilon} \in U_\beta \cdot U_\beta = U_\beta^2 \subMenge U_\alpha }

From ${\textstyle {\frac {x}{\left\|x\right\|_{\beta }+\varepsilon }}\cdot {\frac {y}{\left\|y\right\|_{\beta }+\varepsilon }}\in U_{\alpha }}$ follows for all ${\textstyle \varepsilon >0}$ using property (A3) which applies with the above proof step for toplogical algebras also:

$${\displaystyle {\begin{array}{rcl}\left\|{\frac {x}{\left\|x\right\|_{\beta }+\varepsilon }}\cdot {\frac {y}{\left\|y\right\|_{\beta }+\varepsilon }}\right\|_{\alpha }&\leq &1\Longrightarrow \\\left\|x\cdot y\right\|_{\alpha }&\leq &(\left\|x\right\|_{\beta }+\varepsilon )\cdot (\left\|y\right\|_{\beta }+\varepsilon )\end{array}}}$$

Since ${\textstyle \varepsilon >0}$ was chosen as desired, it is also valid ${\textstyle \left\|x\cdot y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }\cdot \left\|y\right\|_{\beta }}$ for all ${\textstyle x,y\in A}$.

Conversely, if on an algebra a Topologie-producing measurement functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is given that properties ${\textstyle (A1)-(A5)}$ are fulfilled, then the stiffness of the multiplication is shown with respect to ${\textstyle {\mathcal {T}}}$.

If the Topologie-producing measurement functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ meets the condition (A5), the following applies:

$${\displaystyle \displaystyle \forall _{\displaystyle \alpha \in {\mathcal {A}}}\exists {\beta \in {\mathcal {A}}}\forall _{\displaystyle x,y\in A}:\left\|x\cdot y\right\|_{\alpha }\leq \left\|x\right\|_{\beta }\cdot \left\|y\right\|_{\beta }}$$

Conversely, with the same definition of ${\textstyle U_{\alpha }}$ and ${\textstyle U_{\beta }}$ as open single balls ${\textstyle U_{\alpha }:=B_{1}^{\alpha }(0_{A})}$ and ${\textstyle U_{\beta }:=B_{\varepsilon }^{\beta }(0_{A})}$ with open zero environments ${\textstyle 0<\varepsilon <1}$ are obtained. We now show that all ${\textstyle x,y\in U_{\beta }}$ is the product ${\textstyle x\cdot y\in U_{\alpha }}$. In particular, ${\textstyle \varepsilon <1}$ applies Failed to parse (unknown function "\subMenge"): {\textstyle U_\beta := B_\varepsilon^\beta (0_A) \subMenge B_1^\beta (0_A) =: U_\gamma } and thus ${\textstyle \|x\|_{\beta }\geq \|x\|_{\gamma }:=p_{U_{\gamma }}(x)}$ (see connection of Mengeninklusion and the ${\textstyle \leq }$ relation over absorbent Mengen and (702-5761583).

In the return direction of (A5), a circular zero environment ${\textstyle U}$ is obtained for each zero environment ${\textstyle U_{\alpha }}$ from the neighborhood system of circular Mengen with Failed to parse (unknown function "\subMenge"): {\textstyle U_\alpha \subMenge U } , which was produced by a measuring functional ${\textstyle \|\cdot \|_{\alpha }}$ as unit. This applies because the measuring functional system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is Topologie-producing. Sei now ${\textstyle x,y\in U_{\beta }}$

$${\displaystyle 1>\varepsilon ^{2}=\varepsilon \cdot \varepsilon \geq \|x\|_{\beta }\cdot \|y\|_{\beta }{\stackrel {(A5)}{\geq }}\|x\cdot y\|_{\alpha }}$$

This gives ${\textstyle \|x\cdot y\|_{\alpha }<1}$ and thus ${\textstyle x\cdot y\in U_{\alpha }}$.

The rigidity of the multiplication in a Topologie produced by the system ${\textstyle \left\|\cdot \right\|_{\mathcal {A}}}$ is then obtained for any zero environment ${\textstyle U\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$ a circular zero environment Failed to parse (unknown function "\subMenge"): {\textstyle U_\alpha \subMenge U } and with (A5) a ${\textstyle U_{\beta }}$ with

Failed to parse (unknown function "\subMengeeq"): {\displaystyle U_\beta \cdot U_\beta = U_\beta^2 \subMengeeq U_\alpha \subMengeeq U }

For the multiplication to ${\textstyle (A,{\mathcal {T}})}$ stetig.

Overall, it was shown that in topologische Algebren ${\textstyle (A,{\mathcal {T}})}$, the firmness of the Verknüpfungen produced on the algebra can also be demonstrated with the topologische Algebren measurement functional system or, for further statements on topologische Algebren, only with the measurement functional system. ${\textstyle \Box }$

Sei ${\textstyle A}$ an algebra. ${\textstyle A}$ is exactly a house village f'sche topologischen Algebra ${\textstyle (A,{\mathcal {T}})}$ if the Topologie ${\textstyle {\mathcal {T}}}$ can be produced by a 'base-producing (698-1047-1761583604265-341-281)-Gaugefunktional system (698-1047)

• (B1) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}},x\in A}\,\left\|x\right\|_{\alpha }\geq 0}$
• (B2) ${\textstyle \displaystyle \left(\forall _{\alpha \in {\mathcal {A}}}\,\left\|x\right\|_{\alpha }=0\right)\Longrightarrow x=0_{A}}$
• (B3) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}},x\in A,\lambda \in \mathbb {K} }\,\left\|\lambda \cdot x\right\|_{\alpha }=|\lambda |^{p}\cdot \left\|x\right\|_{\alpha }}$
• (B4) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}}}\exists _{\beta \in {\mathcal {A}},K>0}\forall _{x,y\in A}\,\left\|x+y\right\|_{\alpha }\leq K\cdot \left(\left\|x\right\|_{\beta }+\left\|y\right\|_{\beta }\right)}$
• (B5) ${\textstyle \displaystyle \forall _{\alpha \in {\mathcal {A}}}\exists _{\beta \in {\mathcal {A}},M>0}\forall _{\displaystyle x,y\in A}\,\left\|x\cdot y\right\|_{\alpha }\leq M\cdot \left\|x\right\|_{\beta }\cdot \left\|y\right\|_{\beta }}$

One basic measuring functional system has the same function as the use of ${\textstyle \varepsilon }$-Umgebungen in the analysis. The ${\textstyle \varepsilon }$-Umgebungen in ${\textstyle \mathbb {R} }$ gave a base of standard Topologie on ${\textstyle \mathbb {R} }$ which can be expressed as convergence, stiffness and other topological properties. An analogous procedure as in measuring functionals was used.

$${\displaystyle x\in B_{\varepsilon }^{|\cdot |}(x_{0})\Longleftrightarrow |x-x_{0}|<\varepsilon \Longleftrightarrow x\in \,\,]x_{0}-\varepsilon ,x_{0}+\varepsilon [}$$

Show the above properties (B1)-(B2) using the property of the measuring functional system to be basic. For each zero environment ${\textstyle U\in {\mathfrak {U}}_{\mathcal {T}}(0_{A})}$ there is a ${\textstyle \alpha \in {\mathcal {A}}}$ ${\textstyle \varepsilon >0}$ with

Failed to parse (unknown function "\subMengeeq"): {\displaystyle U \subMengeeq B_\varepsilon^{\alpha}(0_A) := \left\{ x \in A \, \colon \, \| x \|_\alpha < \varepsilon \right\} }

Sei ${\textstyle A:={\mathcal {C}}(\mathbb {R} ,\mathbb {R} )}$ Menge the stetig functions of ${\textstyle \mathbb {R} }$ after ${\textstyle \mathbb {R} }$ with the following addition and multiplication on the Vektorraum:

• (Addition) ${\textstyle f,g\in A}$ are defined ${\textstyle f+g}$ argument ${\textstyle (f+g)(x):=f(x)+g(x)}$ for all ${\textstyle x\in \mathbb {R} }$
• (multiplication) ${\textstyle f,g\in A}$ is defined ${\textstyle f\cdot g}$ argument ${\textstyle (f\cdot g)(x):=f(x)\cdot g(x)}$ for all ${\textstyle x\in \mathbb {R} }$

Furthermore, a measuring functional system ${\textstyle \|\cdot \|_{\mathcal {A}}}$ is defined with ${\textstyle {\mathcal {A}}:=\mathbb {R} }$ and ${\textstyle \|f\|_{\alpha }:=|f(\alpha )|}$.

Show that ${\textstyle \|\cdot \|_{\mathcal {A}}}$ is a semi-standard system.

• Specify a Failed to parse (unknown function "\Mengeminus"): {\textstyle f \in A\Mengeminus\{0_A\} } for which ${\textstyle \|f\|_{\alpha }=0}$ applies to ${\textstyle \alpha =2}$ and ${\textstyle \alpha =-2}$.
• Specify Failed to parse (unknown function "\Mengeminus"): {\textstyle g \in A\Mengeminus\{0_A\} } , for which ${\textstyle \|f\|_{\alpha }=0}$ applies ${\textstyle \alpha \in \mathbb {Z} }$.

Sei ${\textstyle K:=\{f\in A\,:\,f{\mbox{ konstant }}\}}$ Menge the constant functions in ${\textstyle A}$, ${\textstyle K_{\varepsilon }^{\alpha }:=\{f\in K\,:\,\|f\|_{\alpha }<\varepsilon \wedge f{\mbox{ konstant }}\}}$ and ${\textstyle N_{\alpha }:=\{f\in K\,:\,\|f\|_{\alpha }=0\}}$. Show that the following statement applies:

$${\displaystyle B_{\varepsilon }^{\alpha }(f):=\{f\}+N_{\alpha }+K_{\varepsilon }^{\alpha }}$$.

Show that the Hausdorff property (A2) applies in this topologischen Algebra. Note that ${\textstyle f,g\in A}$ applies with ${\textstyle f\not =g}$ and you can therefore find a ${\textstyle \alpha _{o}\in {\mathcal {A}}}$ in which ${\textstyle f(\alpha _{o})\not =g(\alpha _{o})}$ applies. Designed with the measuring functionals two neighborhoods ${\textstyle U_{1}\in {\mathfrak {U}}_{\mathcal {T}}(f)}$ and ${\textstyle U_{2}\in {\mathfrak {U}}_{\mathcal {T}}(g)}$ with Failed to parse (unknown function "\emptyMenge"): {\textstyle U_1 \cap U_2 = \emptyMenge } .

Show that the Menge ${\textstyle N_{\alpha }:=\{f\in K\,:\,\|f\|_{\alpha }=0\}}$ is a Ideal in topologischen Algebra ${\textstyle A:={\mathcal {C}}(\mathbb {R} ,\mathbb {R} )}$.

• Minkowski functionals
• measure functional
• Continuity sequence
• Ideal
• Topologische Algebra
• Topologische Gruppe
• Convergence in functional Raums