Generating set of a topological algebra
- Generating set of a topological algebra
A generating set "S" of a topological algebra (e.g., a Banach algebra) "A" is a subset of "A" such that the smallest closed subalgebra of "A" containing "S" is "A" itself.
Since polynomials are dense in the set of continuous functions on the interval ,the set (and any of its supersets) consisting of the function is a generating set of the "Banach algebra" . However, it is not a generating set of the "algebra" (since in the definition of a generating set of an algebra the word "closed" is omitted).
A generating set is sometimes called a system of generators.
A structure (e.g., a topological algebra) "A" is called n-generated if there exists a generating set of "A" consisting of at most n elements. If "A" is n-generated for some finite n (resp., for n=1), then "A" is called finitely generated (resp., singly generated).
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