Free Algebra with Countable Basis

TL;DR

This work explores the structure of D-modules with countable bases, analyzing different basis types like Hamel and Schauder, and their convergence properties, to deepen understanding of their algebraic and topological features.

Contribution

It provides a detailed examination of D-modules with countable bases, comparing algebraic and topological bases and their convergence behaviors.

Findings

Hamel basis analysis of D-modules without topology

Schauder basis with normal convergence in D-modules

Comparison of algebraic and topological basis properties

Abstract

In this book I treat the structure of D-module which has countable basis. If we do not care for topology of D-module, then we consider Hamel basis. If norm is defined in D-module, then we consider Schauder basis. In case of Schauder basis, we consider vectors whose expansion in the basis converges normally.