Inverse-producing extensions of Topological Algebras/Algebra of power series

Introduction

For the multiplicative algebraic closure of an algebra $A$, which contains an additional element $t\notin A$, more elements must added to an algebra:

• the multiplication with itself yield elements $t^n\in A[t]$ with $n\in {\mathbb{N}}_o$, $t^0:=e$) and also
• any multiplications of $t^n\in A[t]$ with elements in $A$ generate elements like $a\cdot t^n\in A[t]$ are back in $A[t]$.
• the additive algebraic closure also requires that polynomials ${\sum }_{k=0}^n{a}_k\cdot t^k\in A[t]$ with coefficients $a_k\in A$ are element of algebraic conclusion.

With a system of topology-producing gauge functionals you can define a topological closure of the polynomial gebra $A[t]$.

Definition: algebra of power series

Be $A^{\mathrm{\infty }}[t]$ the set of all powers series with coefficients in $A$ of the form

$$p(t)=\sum _{k=0}^{\mathrm{\infty }}{p}_k\cdot t^k\text{ mit }(p_k{)}_{k\in {\mathbb{N}}_0}\in A^{{\mathbb{N}}_0}$$

Note

The notation of ${\sum }_{k=0}^{\mathrm{\infty }}\dots$ cannot say anything about the convergence of a series, because a topologisation of the algebra is necessary. $A^{\mathrm{\infty }}[t]$ defines purely algebraic a power series with arbitrary coefficients from the algebra $A$.

Power series as sequence of partial sums

For a fixed $t\in 𝕂$, $p(t)$ is used as the sequence of the partial sums

$$(p^{\downarrow n}(t){)}_{n\in \mathbb{N}}:={(\sum _{k=0}^n{p}_k\cdot t^k)}_{n\in \mathbb{N}}$$

Cauchy product

$A^{\mathrm{\infty }}[t]$, analogously to the polynomial gebra, the cauchy multiplication of two potency rows $p,q$ is defined as multiplicative operation as follows.

$$\begin{array}{ccc}p(t)& :=& \sum _{k=0}^{\mathrm{\infty }}{p}_k\cdot t^k\text{, }q(t):=\sum _{k=0}^{\mathrm{\infty }}{q}_k\cdot t^k\text{ und }\\ (p\cdot q)(t)& :=& p(t)\cdot q(t):=\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^n{p}_k\cdot q_{n-k}\cdot t^n.\end{array}$$

Constants as power series

An element $x\in A$ can be identified with the constant polynomial $x(t):=x\cdot t^0\in A[t]\subseteq A^{\mathrm{\infty }}[t]$.

Equality of power series

Be two power series $p,q\in A^{\mathrm{\infty }}[t]$ given with:

$$p(t):=\sum _{k=0}^{\mathrm{\infty }}{p}_k\cdot t^k\text{ and }q(t):=\sum _{k=0}^{\mathrm{\infty }}{q}_k\cdot t^k$$

The equality of potency rows $p=q$ is defined by the coefficient equality:

$$p=q:⟺p_k=q_k\text{ for all }k\in {\mathbb{N}}_0$$

Comments - Equality

The equality of power series or Polynomials do not necessarily have to be defined by the coefficient equation, but can also be defined by the equality of images $p(t)=q(t)$ for all $t\in 𝔻$ from the definition range $t\in 𝔻$.

Example - Equality

If, for example, the residual class ring $R_3:=\{{\bar{0}}_3,{\bar{1}}_3,{\bar{2}}_3\}$ modulo 3 is used as the definition range of a polynomial, the polynomial differs

$$p(t):=t\cdot (t-{\bar{1}}_3)\cdot (t+{\bar{1}}_3)={\bar{1}}_3\cdot t^3-{\bar{1}}_3\cdot t$$

of the zero polynomial with respect to the coefficients of $t^3$ and $t^1$. Nevertheless, the condition $t\in R_3$ applies to all $p(t)={\bar{0}}_3$.

Use in this learning unit

In the further learning unit on topological invertability criteria, the equality of the power series or Polynomials then and only if two polynomials are coefficient for all coefficients of $t^k$.

Topological algebra of power series

Let $(A,{\Vert \cdot \Vert }_{𝒜})$ be an algebra and $A^{\mathrm{\infty }}[t]$ the algebra of the power series with coefficients in $A$. Furthermore, a system of gauge functional ${\Vert |\cdot |\Vert }_{\widetilde{𝒜}}$ is defined, then the topological closure of the polynomialgebra $\overline{A[t]}$ is designated by $A[t]$. All $p\in A^{\mathrm{\infty }}[t]$ order $\overline{A[t]}$ if the following condition applies ${\Vert \left|p\right|\Vert }_{\tilde{\alpha }}<\mathrm{\infty }$ for all $\tilde{\alpha }\in \widetilde{𝒜}$.

Induced topology from the algebra to the algebra of power series

Be $(A,{\Vert \cdot \Vert }_{𝒜})\in 𝒦(𝕂)$ a topological algebra of class $𝒦$. Furthermore, a positive constant $\alpha \in 𝒜$ and $k\in {\mathbb{N}}_0$ and a $C_k(\alpha )$ and a $𝒦$ functional ${\Vert \cdot \Vert }_k^{(\alpha )}$ are selected, by the following Gauge functionals: $p$-Gaugefunctional on vector space of all Potency rows with coefficients are defined in $A$:

$${\Vert \left|p\right|\Vert }_{\alpha }:=\sum _{k=0}^{\mathrm{\infty }}{C}_k(\alpha )\cdot {\Vert p_k\Vert }_k^{(\alpha )}.$$

Topological closure of the polynomial gebra with respect to the gauge functional system

$(\overline{A[t]},{\Vert |\cdot |\Vert }_{𝒜})$ then refer to the topological conclusion of $A[t]$ with respect to ${\Vert |\cdot |\Vert }_{𝒜}$, i.e. vector space of all potency rows with coefficients in $A$, which additionally meet the following condition:

$${\Vert \left|p\right|\Vert }_{\alpha }<\mathrm{\infty }\text{ für alle }\alpha \in 𝒜.$$

Topologizing of algebra of power series and algebra extension

The algebra of power series $\overline{A[t]}$ is now topologized in a way that depends on the Gauge functional system on $A$. This procedure is necessary in order to embed the algebra $B$ in $A$ in $B$. That is to say the unital algebra $A\in {𝒦}_e$ from a class ${𝒦}_e$ is added to the algebra extension $B\in {𝒦}_e$ by an algebraisomorphism $\tau :A⟶A^{\prime }\subset B$ embedded:

• $\tau (e_{{}_A})=e_{{}_B}$, where $e_{{}_A}$ is the single element of $A$ and $e_{{}_B}\in A^{\prime }$ is the single element of $B$.
• $A$ is homeomorphous to $A^{\prime }$; i.e. $\tau$ and ${\tau }^{-1}:A^{\prime }⟶A$ are steady.

Note: Stetility algebraisomorphism

The rigidity of algebraisomorphism and the reverse image ${\tau }^{-1}:A^{\prime }⟶A$ is later detected by the gauge functional systems on $A\in {𝒦}_e$ and the relative topology induced from $B\in {𝒦}_e$ on $A^{\prime }\in {𝒦}_e$.

Lemma: Isotone sequence of gauge functional

Be $(A,\Vert \cdot {\Vert }_{𝒜})\in 𝒦$ and $({\Vert \cdot \Vert }_n^{(\alpha )}{)}_{n\in \mathbb{N}}$ isotone sequences of Gauge functionals with Coefficients $C_k^n(\alpha )\ge 1$, applicable to:

• ${\Vert \cdot \Vert }_1^{(\alpha )}:={\Vert \cdot \Vert }_{\alpha }$ for all $\alpha \in 𝒜$
• ${\Vert \cdot \Vert }_n^{(\alpha )}\le {\Vert \cdot \Vert }_{n+1}^{(\alpha )}$ for all $n\in \mathbb{N}$ and $\alpha \in 𝒜$
• $C_k^n(\alpha )\le C_k^{n+1}(\alpha )$ for all $\alpha \in 𝒜$ and $n,k\in \mathbb{N}$
• $C_o^n(\alpha )=1$ for all $n\in \mathbb{N}$ and $\alpha \in 𝒜$.

Prerequisite 2 - Gauge-functional systems on algebra of power series

The following four systems are available at $A[t]$: ${\Vert |\cdot |\Vert }_{𝒜\times \mathbb{N}}^{(I)}$, ${\Vert |\cdot |\Vert }_{𝒜\times \mathbb{N}}^{(II)}$, ${\Vert |\cdot |\Vert }_{𝒜\times \mathbb{N}}^{(III)}$, ${\Vert |\cdot |\Vert }_{𝒜\times \mathbb{N}}^{(IV)}$ by Gauge-functional for $p(t)=\sum _{k=0}^{\mathrm{\infty }}{p}_{{}_k}\cdot t^k\in A[t]$ defined:

Prerequisite 2 - Definition of Gauge Functional Systems

${\Vert \left|p\right|\Vert }_{(\alpha ,n)}^{(I)}:=\sum _{k=0}^{\mathrm{\infty }}{C}_k^n(\alpha )\cdot {\Vert p_{{}_k}\Vert }_n^{(\alpha )}$ ${\Vert \left|p\right|\Vert }_{(\alpha ,n)}^{(II)}:=\sum _{k=0}^{\mathrm{\infty }}{C}_k^n(\alpha )\cdot {\Vert p_{{}_k}\Vert }_{n+1}^{(\alpha )}$ ${\Vert \left|p\right|\Vert }_{(\alpha ,n)}^{(III)}:=\sum _{k=0}^{\mathrm{\infty }}{C}_k^{n+1}(\alpha )\cdot {\Vert p_{{}_k}\Vert }_n^{(\alpha )}$ ${\Vert \left|p\right|\Vert }_{(\alpha ,n)}^{(IV)}:={\Vert p_o\Vert }_n^{(\alpha )}+\sum _{k=1}^{\mathrm{\infty }}{C}_k^n(\alpha )\cdot {\Vert p_{{}_k}\Vert }_{n+2}^{(\alpha )}$

sequencerung - topology generation

With the above conditions, the systems generate the same on $A[t]$ topology. In particular, a solid $\alpha$ is obtained for all 4 selected subsystems of gauge functional ${\left|\Vert \cdot \Vert \right|}_{\{\alpha \}\times \mathbb{N}}^{(I)}$,\dots , ${\left|\Vert \cdot \Vert \right|}_{\{\alpha \}\times \mathbb{N}}^{(IV)}$. the same subsystem ${𝒯}_{\alpha }$ open sets of topology $𝒯$.

Proof

All $\alpha \in 𝒜$, $n\in \mathbb{N}$ and $M\in \{I,\dots ,IV\}$ the following inequality chain:

$${\left|\Vert \cdot \Vert \right|}_{(\alpha ,n)}^{(I)}\le {\left|\Vert \cdot \Vert \right|}_{(\alpha ,n)}^{(M)}\le {\left|\Vert \cdot \Vert \right|}_{(\alpha ,n+2)}^{(I)}$$

The subsystems thus agree with a fixed $\alpha$, i.e. Output topology in accordance with q.e.d.

Equality of partial sums of potency rows

The coefficients of the elements of $A^{\mathrm{\infty }}[t]$ can also be determined clearly via the partial sums. The partial sums are clearly defined as linear combinations in $A$ with $t\in 𝕂$. However, the partial sums as sequence in $A$ do not necessarily have to converge.

Coefficient comparison for partial sums 1

$$\begin{array}{ccc}p(t):=\sum _{k=0}^{\mathrm{\infty }}{p}_k\cdot t^k& ,& q(t):=\sum _{k=0}^{\mathrm{\infty }}{q}_k\cdot t^k\in A[t]\\ & & \text{ mit }{p}^{\downarrow m}(t)=q^{\downarrow m}(t)\mathrm{\forall }t\in 𝕂\\ & ⟹& 0_A=\sum _{k=0}^{\mathrm{\infty }}(p_k-q_k)t^k\end{array}$$

Coefficient comparison for partial sums 2

$$\begin{array}{ccc}& ⟹& {\mathrm{\forall }}_{\alpha \in 𝒜;n\in \mathbb{N}}:0={\left|\Vert p-q\Vert \right|}_{(\alpha ,n)}=\sum _{k=0}^{\mathrm{\infty }}{\Vert p_k-q_k\Vert }_n^{(\alpha )}\\ & ⟹& {\mathrm{\forall }}_{\alpha \in 𝒜,n\in \mathbb{N}}:{\Vert p_k-q_k\Vert }_n^{(\alpha )}=0.\end{array}$$

Since $A$ is a house village flock, $p_k=q_k$ applies to all $k\in {\mathbb{N}}_0$.

Definition: algebra of power series

Be $A$ an algebra and $A^{\mathrm{\infty }}[t]$ the algebra of all potency rows with coefficients in $A$ with Cauchymultiplication. Partial sum up to grade $m\in {\mathbb{N}}_0$ Potency series $p(t)={\sum }_{k=0}^{\mathrm{\infty }}{p}_k\cdot t^k\in \overline{A[t]}$ is the following polynomial:

$$p^{\downarrow m}(t):=\sum _{k=0}^m{p}_k\cdot t^k.$$

Definition: Partial Sum Topology

Be $(A[t],{\left|\Vert \cdot \Vert \right|}_{𝒜})$ a polynomial gebra. Reference ${\left|\Vert \cdot \Vert \right|}_{𝒜}^{\downarrow \mathbb{N}}$ the system of partial sum functions of ${\left|\Vert \cdot \Vert \right|}_{𝒜}$

$$\begin{array}{ccc}p(t)& :=& \sum _{k=0}^{\mathrm{\infty }}{p}_k\cdot t^k\text{wie folgt definiert sind:}\\ {\left|\Vert p\Vert \right|}_{\alpha }^{\downarrow m}& :=& {\left|\Vert p^{\downarrow m}\Vert \right|}_{\alpha }\text{ mit }\alpha \in 𝒜\text{ und }m\in \mathbb{N}.\end{array}$$

The ${\left|\Vert \cdot \Vert \right|}_{𝒜}^{\downarrow \mathbb{N}}$ generated topology is called partial sum topology of ${\left|\Vert \cdot \Vert \right|}_{𝒜}$ on $A[t]$.

Note

The partial sum topology is more than that of ${\left|\Vert \cdot \Vert \right|}_{𝒜}$ produced starting topology, for for $\alpha \in 𝒜$:

$${\left|\Vert \cdot \Vert \right|}_{\alpha }^{\downarrow m}\le {\left|\Vert \cdot \Vert \right|}_{\alpha }\text{ für alle }m\in \mathbb{N}.$$

The partial sum topology is obtained by individual gauge functional ${\left|\Vert \cdot \Vert \right|}_{𝒜}$ with Projections ${\cdot }^{\downarrow m}:A[t]⟶A[t]$ linked to the first $m$ summands of the polynomial and as selects topology-producing functionals on $A[t]$. $m\in \mathbb{N}$ selected as desired. The following Lemma shows the clarity of the factorization of any desired Items $p\in \overline{A[t]}$ by $z\cdot t-e\in A[t]$ and one by $p$ selected formal potency series $\hat{p}\in \overline{A[t]}$.

Tasks

In the following tasks, some small exercises will be used to calculate

Norm - Matrixalegbra - Topologising algebra of power series

The two dies are given

$$r_k:=\left(\begin{array}{cc}\frac{1}{2^k}& \frac{1}{3^k}\\ \frac{1}{4^k}& \frac{1}{5^k}\end{array}\right)e_A:=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$

with the single element $e_A$ in the algebra $A:=Mat(2\times 2,ℝ)$. $A$ is standard

$$\Vert a{\Vert }_{max}={\Vert \left(\begin{array}{cc}{a}_1& a_2\\ a_3& a_4\end{array}\right)\Vert }_{max}=\max \{|a_1|,|a_2|,|a_3|,|a_4|\}$$

a normed space.

• Show that the potency series $q(t):=\sum _{k=0}^{\mathrm{\infty }}{e}_A\cdot t^k$ and the standard $\Vert |p|\Vert :=\sum _{k=0}^{\mathrm{\infty }}\Vert p_k{\Vert }_{max}$ are not in $\overline{A[t]}$.
• Calculate $\Vert |q|\Vert$ and $\Vert |r|\Vert$ with $q_k=e_A$ and the above-defined coefficients in$r_k$.
• Calculate the matrix $r\in \overline{A[t]}$ with $t=1$!

Vector space of real-value continuous functions

We use as domain and range $ℝ$, which the algebra $A:=𝒞(ℝ,ℝ)$ of the continuous functions of $ℝ$ to $ℝ$ with the seminorms

$$\Vert f{\Vert }_n:=\underset{x\in [-n,+n]}{\max }|f(x)|$$

becomes $(A,\Vert \cdot {\Vert }_{\mathbb{N}})$ a local convex topological vector space.

• Topologize the polynomial agebra $A[t]$ with a system of seminorms $(A[t],\Vert |\cdot |{\Vert }_{\mathbb{N}})$, which is defined via the seminorms $\Vert \cdot {\Vert }_n$ on $A$.

$\Vert |p|{\Vert }_n:=\sum _{k=0}^n{C}_k\cdot \Vert p_k{\Vert }_n$.

select the coefficients as a geometric series for the $(C_k{)}_{k\in \mathbb{N}}$

• Show that the seminorms $\Vert \cdot {\Vert }_n$ are submultiplicative, i.e. $\Vert f\cdot g{\Vert }_n\le \Vert f{\Vert }_n\cdot \Vert g{\Vert }_n$!
• Select the coefficients $C_k\in {ℝ}_o^{+}$ so that the polynomial $q\in A[t]$ with $q_k=cos$ as the cosine function for all $k\in {\mathbb{N}}_o$ is an element of the algebra of power series $A^{\mathrm{\infty }}[t]$ The object $q$ is thus a power series with coefficients $q_k$ in $A$ as cosine function. For example, select $C_k:=\frac{1}{3^k}$ and calculate $\Vert cos{\Vert }_n$ for all $n\in \mathbb{N}$. The sequence of coefficients $(C_k{)}_{k\in {\mathbb{N}}_0}$ must have a general characteristic for providing $\Vert |q|{\Vert }_n<\mathrm{\infty }$ for all $n\in \mathbb{N}$, i.e. for all $n$ the seminorm must yield a finite value.
• Choose a geometric series of $(C_k{)}_{k\in \mathbb{N}}$ with $0<q<1$ and $C_k:=q^k$ and show that

$$\Vert |p\cdot q|{\Vert }_n\le \Vert |p|{\Vert }_n\cdot \Vert |q|{\Vert }_n$$

with the Cauchy product on $A[t]$.

Factorization problem for zt-e

Be selected as desired $A$ a unital algebra with single element $e$ and $z\in A$. $A^{\mathrm{\infty }}[t]$ is with the cauchy multiplication an algebra in which:

$${\mathrm{\forall }}_{p\in A^{\mathrm{\infty }}[t]}{\mathrm{\exists }}_{\hat{p}\in A^{\mathrm{\infty }}[t]}:p(t)=(z\cdot t-e)\cdot \hat{p}(t)(\ast )$$

$\hat{p}$ is clearly defined for each $p$.

Proof

$A^{\mathrm{\infty }}[t]$ is a unital ring and $s\in A[t]\subset A^{\mathrm{\infty }}[t]$ with $s(t):=z\cdot t-e=z\cdot t^1+e\cdot t^0$. We now show that $s\in A^{\mathrm{\infty }}[t]$ can be inverted.

Inverse element of zt-e

A polynomial $z\in A$ is first defined using the given $q\in A^{\mathrm{\infty }}[t]$:

$$q(t):=-\sum _{k=0}^{\mathrm{\infty }}{z}^k\cdot t^k=\sum _{k=0}^{\mathrm{\infty }}-z^k\cdot t^k$$

We now calculate $s\cdot q\in A^{\mathrm{\infty }}[t]$

$$\begin{array}{ccc}(s\cdot q)(t)& =& s(t)\cdot q(t)=(z\cdot t-e)\cdot (-\sum _{k=0}^{\mathrm{\infty }}{z}^k\cdot t^k)\\ & =& -\sum _{k=1}^{\mathrm{\infty }}{z}^k\cdot t^k+\sum _{k=0}^{\mathrm{\infty }}{z}^k\cdot t^k=e\end{array}$$

Definition of the searched power series

This defines $\hat{p}(t):=q(t)\cdot p(t)=(z\cdot t-e{)}^{-1}\cdot p(t)$.

Unambiguousness of the power series

Uniqueness of $\hat{p}$: Be given $\widehat{p_{(1)}},\widehat{p_{(2)}}\in \overline{A[t]}$ which possess the property $(*)$. For $v(t):=z\cdot t-e$ is obtained:

$$\begin{array}{ccc}p& =& v\cdot \widehat{p_{(1)}}=v\cdot \widehat{p_{(2)}}⟹\\ \widehat{p_{(1)}}& =& v^{-1}\cdot (v\cdot \widehat{p_{(1)}})=v^{-1}\cdot (v\cdot \widehat{p_{(2)}})=\widehat{p_{(2)}}\end{array}$$

q.e.d.

Note

The coefficients of the elements of $A^{\mathrm{\infty }}[t]$ are clearly defined, unless:

$$\begin{array}{ccc}p(t):=\sum _{k=0}^{\mathrm{\infty }}{p}_k\cdot t^k& ,& q(t):=\sum _{k=0}^{\mathrm{\infty }}{q}_k\cdot t^k\in A[t]\text{ mit }p(t)=q(t)\\ & ⟹& 0\equiv \sum _{k=0}^{\mathrm{\infty }}(p_k-q_k)t^k\\ & ⟹& {\mathrm{\forall }}_{\alpha \in 𝒜;n\in \mathbb{N}}:0={\Vert |p-q|\Vert }_{(\alpha ,n)}=\sum _{k=0}^{\mathrm{\infty }}{\Vert p_k-q_k\Vert }_n^{(\alpha )}\\ & ⟹& {\mathrm{\forall }}_{\alpha \in 𝒜,n\in \mathbb{N}}:{\Vert p_k-q_k\Vert }_n^{(\alpha )}=0.\end{array}$$

Since $A$ is a house village flock, $p_k=q_k$ applies to all $k\in {\mathbb{N}}_0$.

Note

The partial total topology is coarser than the starting topology produced by ${\Vert |\cdot |\Vert }_{𝒜}$, since $\alpha \in 𝒜$ applies:

$${\Vert |\cdot |\Vert }_{\alpha }^{\downarrow m}\le {\Vert |\cdot |\Vert }_{\alpha }\text{ für alle }m\in \mathbb{N}.$$

The partial sum topology is obtained by individual gauge functional ${\Vert |\cdot |\Vert }_{𝒜}$ with Projections ${\cdot }^{\downarrow m}:A[t]⟶A[t]$ linked to the first $m$ summands of the polynomial and as selects topology-producing functionals on $A[t]$. $m\in \mathbb{N}$ selected as desired. The following Lemma shows the clarity of the factorization of any desired Items $p\in \overline{A[t]}$ by $z\cdot t-e\in A[t]$ and one by $p$ selected formal potency series $\hat{p}\in \overline{A[t]}$.

See also

• Algebra of polynomial
• Multiplicative topological devisors of zero

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