Infinite Dimensional Vector Space
Published on Jun 15, 2023
We are familiar with the properties of finite dimensional vector spaces over a field. Many of the results that are valid in finite dimensional vector spaces can very well be extended to infinite dimensional cases sometimes with slight modifications in definitions. But there are certain results that do not hold in infinite dimensional cases. Here we consolidate some of those results and present it in a readable form.
We present the whole work in three chapters. All those concepts in vector spaces and linear algebra which we require in the sequel are included in the first chapter. In section I of chapter II we discuss the fundamental concepts and properties of infinite dimensional vector spaces and in section II, the properties of the subspaces of infinite dimensional vector spaces are studied and will find that the chain conditions which hold for finite cases do not hold for infinite cases.
The linear transformation on infinite dimensional vector spaces and introduce the concept of infinite matrices. We will show that every linear transformation corresponds to a row finite matrix over the underlying field and vice versa and will prove that the set of all linear transformations of an infinite dimensional vector space in to another is isomorphic to the space of all row finite matrices over the underlying field.
In section II we consider the conjugate space of an infinite dimensional vector space and define its dimension and cardinality and will show that the dimension of the conjugate space is greater than the original space. Finally we will show that the conjugate space of the conjugate space of an infinite dimensional vector space cannot be identified with the original space.
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