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

Introduction

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For the multiplicative algebraic closure of an algebra  , which contains an additional element  , more elements must added to an algebra:

  • the multiplication with itself yield elements   with  ,  ) and also
  • any multiplications of   with elements in   generate elements like   are back in  .
  • the additive algebraic closure also requires that polynomials   with coefficients   are element of algebraic conclusion.

With a system of topology-producing gauge functionals you can define a topological closure of the polynomial gebra  .

Definition: algebra of power series

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Be   the set of all powers series with coefficients in   of the form

 

Note

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The notation of   cannot say anything about the convergence of a series, because a topologisation of the algebra is necessary.   defines purely algebraic a power series with arbitrary coefficients from the algebra  .

Power series as sequence of partial sums

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For a fixed  ,   is used as the sequence of the partial sums

 

Cauchy product

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 , analogously to the polynomial gebra, the cauchy multiplication of two potency rows   is defined as multiplicative operation as follows.

 

Constants as power series

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An element   can be identified with the constant polynomial  .

Equality of power series

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Be two power series   given with:

 

The equality of potency rows   is defined by the coefficient equality:

 

Comments - Equality

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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   for all   from the definition range  .

Example - Equality

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If, for example, the residual class ring   modulo 3 is used as the definition range of a polynomial, the polynomial differs

 

of the zero polynomial with respect to the coefficients of   and  . Nevertheless, the condition   applies to all  .

Use in this learning unit

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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  .

Topological algebra of power series

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Let   be an algebra and   the algebra of the power series with coefficients in  . Furthermore, a system of gauge functional   is defined, then the topological closure of the polynomialgebra   is designated by  . All   order   if the following condition applies   for all  .

Induced topology from the algebra to the algebra of power series

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Be   a topological algebra of class  . Furthermore, a positive constant   and   and a   and a   functional   are selected, by the following Gauge functionals:  -Gaugefunctional on vector space of all Potency rows with coefficients are defined in  :

 

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

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  then refer to the topological conclusion of   with respect to  , i.e. vector space of all potency rows with coefficients in  , which additionally meet the following condition:

 

Topologizing of algebra of power series and algebra extension

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The algebra of power series   is now topologized in a way that depends on the Gauge functional system on  . This procedure is necessary in order to embed the algebra   in   in  . That is to say the unital algebra   from a class   is added to the algebra extension   by an algebraisomorphism   embedded:

  •  , where   is the single element of   and   is the single element of  .
  •   is homeomorphous to  ; i.e.   and   are steady.

Note: Stetility algebraisomorphism

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The rigidity of algebraisomorphism and the reverse image   is later detected by the gauge functional systems on   and the relative topology induced from   on  .

Lemma: Isotone sequence of gauge functional

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Be   and   isotone sequences of Gauge functionals with Coefficients  , applicable to:

  •   for all  
  •   for all   and  
  •   for all   and  
  •   for all   and  .

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

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The following four systems are available at  :  ,  ,  ,   by Gauge-functional for   defined:

Prerequisite 2 - Definition of Gauge Functional Systems

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sequencerung - topology generation

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With the above conditions, the systems generate the same on   topology. In particular, a solid   is obtained for all 4 selected subsystems of gauge functional  ,\dots ,  . the same subsystem   open sets of topology  .

Proof

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All  ,   and   the following inequality chain:

 

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

Equality of partial sums of potency rows

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The coefficients of the elements of   can also be determined clearly via the partial sums. The partial sums are clearly defined as linear combinations in   with  . However, the partial sums as sequence in   do not necessarily have to converge.

Coefficient comparison for partial sums 1

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Coefficient comparison for partial sums 2

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Since   is a house village flock,   applies to all  .

Definition: algebra of power series

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Be   an algebra and   the algebra of all potency rows with coefficients in   with Cauchymultiplication. Partial sum up to grade   Potency series   is the following polynomial:

 

Definition: Partial Sum Topology

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Be   a polynomial gebra. Reference   the system of partial sum functions of  

 

The   generated topology is called partial sum topology of   on  .

Note

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The partial sum topology is more than that of   produced starting topology, for for  :

 

The partial sum topology is obtained by individual gauge functional   with Projections   linked to the first   summands of the polynomial and as selects topology-producing functionals on  .   selected as desired. The following Lemma shows the clarity of the factorization of any desired Items   by   and one by   selected formal potency series  .

Tasks

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

Norm - Matrixalegbra - Topologising algebra of power series

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The two dies are given

 

with the single element   in the algebra  .   is standard

 

a normed space.

  • Show that the potency series   and the standard   are not in  .
  • Calculate   and   with   and the above-defined coefficients in .
  • Calculate the matrix   with  !

Vector space of real-value continuous functions

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We use as domain and range  , which the algebra   of the continuous functions of   to   with the seminorms

 

becomes   a local convex topological vector space.

  • Topologize the polynomial agebra   with a system of seminorms  , which is defined via the seminorms   on  .
 .
select the coefficients as a geometric series for the  
  • Show that the seminorms   are submultiplicative, i.e.  !
  • Select the coefficients   so that the polynomial   with   as the cosine function for all   is an element of the algebra of power series   The object   is thus a power series with coefficients   in   as cosine function. For example, select   and calculate   for all  . The sequence of coefficients   must have a general characteristic for providing   for all  , i.e. for all   the seminorm must yield a finite value.
  • Choose a geometric series of   with   and   and show that
 
with the Cauchy product on  .

Factorization problem for zt-e

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Be selected as desired   a unital algebra with single element   and  .   is with the cauchy multiplication an algebra in which:

 

  is clearly defined for each  .

Proof

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  is a unital ring and   with  . We now show that   can be inverted.

Inverse element of zt-e

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A polynomial   is first defined using the given  :

 

We now calculate  

 

Definition of the searched power series

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This defines  .

Unambiguousness of the power series

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Uniqueness of  : Be given   which possess the property  . For   is obtained:

 

q.e.d.

Note

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The coefficients of the elements of   are clearly defined, unless:

 

Since   is a house village flock,   applies to all  .

Note

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The partial total topology is coarser than the starting topology produced by  , since   applies:

 

The partial sum topology is obtained by individual gauge functional   with Projections   linked to the first   summands of the polynomial and as selects topology-producing functionals on  .   selected as desired. The following Lemma shows the clarity of the factorization of any desired Items   by   and one by   selected formal potency series  .

See also

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