Inverse-producing extensions of Topological Algebras/topological algebra

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Definition: Topological Vector Space[edit | edit source]

A topological vector space over is a vector space over the body that has a topology with which scalar multiplication and addition are continuous mappings.

In the following, for all topological vector spaces, we shall use the Hausdorff property be assumed.

Definition: Neighbourhood[edit | edit source]

Let be a topological space with a topology as a system of open sets and , then denote

  • the set of all neighbourhoods from the point ,
  • the set of all open Neighbourhoods from the point ,
  • the set of all closed neighbourhoods of point .

Remark: Indexing with topology[edit | edit source]

If no misunderstanding about the underlying topological space can occur, the index is not included as a designation of the topology used.

Remark: Analogy to the epsilon neighbourhood[edit | edit source]

In convergence statements in the real numbers one usually considers only neighbourhood. In doing so, one would actually have to consider in topological spaces for arbitrary neighbourhoods from find an index bound of a net above which all lie with . However, since the neighbourhoods are an neighbourhood basis, by the convergence definition one only needs to show the property for all neighbourhoods with .

Convergence in topological spaces[edit | edit source]

Let be a topological space, , an index set (partial order) and a mesh. The convergence of against is then defined as follows:


(where "" for is the partial order on the index set).

Definiton: Neighbourhood basis[edit | edit source]

Let be a topological space, and the set of all neighbourhoods of . is called the neighbourhood basis of if for every :.

Remark: Epsilon spheres in normalized spaces[edit | edit source]

Let be a normed space, then the spheres form

an ambient basis of the set of all environments of of .

Learning Task 1[edit | edit source]

Let be a toplogic space with chaotic topology .

  • Determine for any .
  • Show that any sequence converges in against any limit .

Learning Task 2[edit | edit source]

Let be a metric space with the discrete topology given by the metric:

  • Determine for any .
  • How many sets make up minimal for any ?
  • Formally state all sequences in that converge to a limit !

Definition: open sets[edit | edit source]

Let be a topological space and be the system of open sets, that is:


Task[edit | edit source]

Let be a topological space on the basic set of real numbers. However, the topology does not correspond to the Euclic topology over the set , but the open sets are defined as follows.

  • Show that is a topological space.
  • Show that the sequence does not converge to in the topological space .

Here is the complement of in .

Remark: open - closed[edit | edit source]

By the system of open sets in a topology the closed sets of the topology are also defined at the same time as their complements.

Definition: closed sets[edit | edit source]

Let be a topological space and be the system of open sets.

Definition: open kernel[edit | edit source]

Let be a topological space and , then the open kernel of is the union of all open subsets of .


Definition: closed hull[edit | edit source]

Let be a topological space. The closed hull of is the intersection over all closed subsets of containing and is open.

Definition: edge of a set[edit | edit source]

The topological edge of is defined as follows:

Remark: sequences and nets[edit | edit source]

In metric spaces, one can still work with the natural numbers as countable index sets. In arbitrary topological spaces one has to generalize the notion of sequences to the notion of nets.

Definition: nets[edit | edit source]

Let be a topological space and an index set (with partial order), then denotes the set of all families indexed by in :

Definition: finite sequences[edit | edit source]

Let be a vector space, then denotes the set of all finite sequences with elements in :

Definition: Algebra[edit | edit source]

An algebra over the body is a vector space over in which a multiplication is an inner join

is defined where for all and the following properties are satisfied:

Definition: topological algebra[edit | edit source]

A topological algebra over the body is a topological vector space over , where also multiplication is

is a continuous inner knotting.

Continuity of multiplication[edit | edit source]

Continuity of multiplication means here:

Multiplicative topology - continuity[edit | edit source]

The topology is called multiplicative if holds:

Remark: Multiplicative topology - Gaugefunctionals[edit | edit source]

In describing topology, the Topologization_lemma for algebras shows that the topology can also be described by a system of Gaugefunctionals

Unitary algebra[edit | edit source]

The algebra is called unital if it has a neutral element of multiplication. In particular, one defines for all . The set of all invertible (regular) elements is denoted by . Non-invertible elements are called singular.

Task: matrix algebras[edit | edit source]

Consider the set of square matrices with matrix multiplication and the maxmum norm of the components of the matrix. Try to prove individual properties of an algebra ( is a non-commutative unitary algebra). For the proof that with matrix multiplication is also a topological algebra, see Topologization Lemma for Algebras.

Definition: sets and links[edit | edit source]

Let be a topological algebra over the body , and be subsets of , then define

Learning Tasks[edit | edit source]

Draw the following set of vectors as sets of points in the Cartesian coordinate system with and and the following intervals :

  • .
  • .
  • .

See also[edit | edit source]

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