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

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

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

An element
can be identified with the constant polynomial
.
Be two power series
given with:

The equality of potency rows
is defined by the coefficient 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
for all
from the definition range
.
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
.
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
.
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.
The rigidity of algebraisomorphism and the reverse image
is later detected by the gauge functional systems on
and the relative topology induced from
on
.
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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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
.
All
,
and
the following inequality chain:

The subsystems thus agree with a fixed
, i.e.
Output topology in accordance with q.e.d.
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.
![{\displaystyle {\begin{array}{rll}p(t):=\displaystyle \sum _{k=0}^{\infty }p_{k}\cdot t^{k}&,&q(t):=\displaystyle \sum _{k=0}^{\infty }q_{k}\cdot t^{k}\in A[t]\\&&{\mbox{ mit }}p^{\downarrow m}(t)=q^{\downarrow m}(t){\mbox{ }}\forall t\in \mathbb {K} \\&\Longrightarrow &0_{A}=\displaystyle \sum _{k=0}^{\infty }(p_{k}-q_{k})t^{k}\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38725a47c3192b7175cc89e8073d506f33154dc4)

Since
is a house village flock,
applies to all
.
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:

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

The
generated topology is called partial sum topology of
on
.
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
.
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
!
We use as domain and range
, which the algebra
of the continuous functions of
to
with the seminorms
![{\displaystyle \|f\|_{n}:=\displaystyle \max _{x\in [-n,+n]}|f(x)|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/364f54b46c4429f921682b8498a86c67594b40f9)
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
.
Be selected as desired
a unital algebra with single element
and
.
is with the cauchy multiplication an algebra in which:
![{\displaystyle \forall _{\displaystyle p\in A^{\infty }[t]}\exists _{\displaystyle {\widehat {p}}\in A^{\infty }[t]}\,:\,p(t)=(z\cdot t-e)\cdot {\widehat {p}}(t)(\ast )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d85ce2785ddfe74dc94d2ab610ede2d6758a82)
is clearly defined for each
.
is a unital ring and
with
. We now show that
can be inverted.
A polynomial
is first defined using the given
:

We now calculate

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

q.e.d.
The coefficients of the elements of
are clearly defined, unless:
![{\displaystyle {\begin{array}{rcl}p(t):=\displaystyle \sum _{k=0}^{\infty }p_{k}\cdot t^{k}&,&q(t):=\displaystyle \sum _{k=0}^{\infty }q_{k}\cdot t^{k}\in A[t]{\mbox{ mit }}p(t)=q(t)\\&\Longrightarrow &0\equiv \displaystyle \sum _{k=0}^{\infty }(p_{k}-q_{k})t^{k}\\&\Longrightarrow &\forall _{\displaystyle \alpha \in {\mathcal {A}};n\in \mathbb {N} }:0=\left\|\!\left|p-q\right|\!\right\|_{(\alpha ,n)}=\sum _{k=0}^{\infty }\left\|p_{k}-q_{k}\right\|_{n}^{(\alpha )}\\&\Longrightarrow &\forall _{\displaystyle \alpha \in {\mathcal {A}},n\in \mathbb {N} }:\left\|p_{k}-q_{k}\right\|_{n}^{(\alpha )}=0.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/317f490ed55f5e4667a287a06a8d555c069390d3)
Since
is a house village flock,
applies to all
.
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
.