In this unit, we introduce the concept of metric spaces. We give the
properties of metric spaces and some theorems related to metric space.
Finally, we explain the Metric Completion Theorem.

Linear space or vector space is the most fundamental and important
structure in the functional analysis. This structure is built making use of
a nonempty set of vectors X , a field of scalars F, which is either field of real numbers or field of complex numbers and two algebraic operations in the form of linear expressions namely, vector addition and scalar multiplication.

In this unit, we shall prove the famous Hahn - Banach Extension Theorem.
This is one of the most fundamental theorems in functional analysis and is due to Hahn and Banach. It yields the existence of nontrivial continuous linear functionals on a normed linear space, '" basic result necessary for the development of a large portion of functional analysis.

In this unit, we introduce the concept of orthogonality and orthogonal complements. We give the properties of orthogonality and orthogonal complements and some theorems related to orthogonality and
orthogonal complements.