Inverse-producing extensions of Topological Algebras/Algebra of power series
Introduction
editFor 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
editBe the set of all powers series with coefficients in of the form
Note
editThe 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
editFor a fixed , is used as the sequence of the partial sums
Cauchy product
edit, analogously to the polynomial gebra, the cauchy multiplication of two potency rows is defined as multiplicative operation as follows.
Constants as power series
editAn element can be identified with the constant polynomial .
Equality of power series
editBe two power series given with:
The equality of potency rows is defined by the coefficient equality:
Comments - Equality
editThe 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
editIf, 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
editIn 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
editLet 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
editBe 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
editthen 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
editThe 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
editThe 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
editBe 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
editThe following four systems are available at : , , , by Gauge-functional for defined:
Prerequisite 2 - Definition of Gauge Functional Systems
edit
sequencerung - topology generation
editWith 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
editAll , 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
editThe 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
editCoefficient comparison for partial sums 2
editSince is a house village flock, applies to all .
Definition: algebra of power series
editBe 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
editBe a polynomial gebra. Reference the system of partial sum functions of
The generated topology is called partial sum topology of on .
Note
editThe 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
editIn the following tasks, some small exercises will be used to calculate
Norm - Matrixalegbra - Topologising algebra of power series
editThe 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
editWe 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
editBe 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
editis a unital ring and with . We now show that can be inverted.
Inverse element of zt-e
editA polynomial is first defined using the given :
We now calculate
Definition of the searched power series
editThis defines .
Unambiguousness of the power series
editUniqueness of : Be given which possess the property . For is obtained:
q.e.d.
Note
editThe coefficients of the elements of are clearly defined, unless:
Since is a house village flock, applies to all .
Note
editThe 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 .