Inverse-producing extensions of Topological Algebras/circular set

Definition: Circular Set

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Let   a vector space over  , then   is circular, if and only if for all   and for all   also   is valid.

Lemma: neighborhood base of the zero vector with circular sets

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In a topological   vector space   there is a neighborhood base of zero vector consisting of balanced (circular) sets.

Proof

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Be   as desired. A   and a zero environment   with

 

with  . The quantity   is circular.

Proof by contradiction

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We now show that   is also a zero environment in  . The assumption is that   is not a zero environment.   is without restriction.

Proof 1: existence of a network

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If   is not a zero environment, there is a network   which is converged against the zero vector  

Proof 2: Convergence against zero vector

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If a network   is converged against the zero vector  , there is also an index barrier   " " means the partial order on the index quantity  .

Proof 3: scalar multiplication for convergent networks

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If a network   is converged against the zero vector  , it also converges   because of the stiffness of the multiplication with scalers in a topological vector space against the zero vector.

Proof 4: scalar multiplication for convergent networks

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Define now a network   with   for all  , which after proof step 3 also against the zero vector   Then there is again an index cabinet  , for which all   are valid if   applies. Here too, " " refers to the partial order on the index quantity  .

Proof 5: contradiction

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Select   in the index quantity such that   and  . For all   the following applies with proof step 1, 4 and  :

  •  
  •  .

Proof 4: circular zero environment

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  is also a circular zero environment and any environment   contains a circular zero environment   with  . The quantity   is a zero environmental base of circular quantities.  


Remark: circular zero environment base

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With this statement, a zero-environmental basis of circular quantities exists in each topological vector space.

Cut circular zero environments

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In topological vector spaces (and thus also topological algebras), it is shown that there is a zero environmental base   of circular quantities  . The circular configuration provides the absolute homogeneity of the Gaugefunctional.

Lemma: Cut circular zero environments

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Be   circular zero environments in a topological vector space  , then also   is a balanced neighborhood of zero.

Proof

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from   follows circularly, for all   with  ,   and  

Intersection of open sets

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In a topological space (in particular also in a topological vector space)  , the intersection of two open quantities is again open, i.e.   (see Norms, metrics, topology).   are neighborhood of the zero vector, then   is valid. Thus,   is an open set containing the zero vector and this yields  .

Intersection circular

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We now show that   is circular. Be selected as   and   with  . This applies to   and  . The circularity of   and   then supplies   and   and thus also  .

Learning Tasks

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  • For the definition of  , show that the set   is circular.
  • Check if the sum   of two circular neighborhoods of the zero vector   is again a circular neighborhoods of the zero vector.
  • Check if the union   of two circular neighborhoods of the zero vector   are circular again

See also

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