| Infinite
Dimensional Vector Space |
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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