Locally Pseudo-Convex Inductive Limit of Topological Algebras
Summary
This article provides an overview of locally pseudo-convex inductive limits of topological algebras. It introduces the concept of a locally pseudo-convex algebra and explains how the inductive limit of a sequence of locally pseudo-convex algebras is also locally pseudo-convex.
Locally Pseudo-Convex Algebras
A locally pseudo-convex algebra is a topological algebra that has a locally pseudo-norm topology. A locally pseudo-norm topology is a topology that is generated by a family of pseudo-seminorms, which are functions that satisfy certain properties similar to norms.
Inductive Limit of Algebras
An inductive limit of a sequence of algebras is a new algebra that is constructed by taking the direct limit of the sequence of algebras. The direct limit is the set of all sequences of elements from the algebras in the sequence that satisfy certain compatibility conditions.
Locally Pseudo-Convex Inductive Limit
The inductive limit of a sequence of locally pseudo-convex algebras is also locally pseudo-convex. This is because the locally pseudo-norm topology on the inductive limit is generated by the family of pseudo-seminorms on the algebras in the sequence.
Applications
Locally pseudo-convex inductive limits of topological algebras have applications in various areas of mathematics, including:
- Holomorphic function theory
- Algebraic geometry
- Differential equations
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