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.