Module 2 : Fundamentals of Vector Spaces
Section 3 : Normed Linear Spaces and Banach Spaces
 

Definition 34 (Banach Space):
A normed linear space $X$ is said to be complete if every Cauchy sequence has a limit in $X$. A complete normed linear space is called Banach space.

Examples of Banach spaces are MATHMATH Concept of Banach spaces can be better understood if we consider an example of a vector space where a Cauchy sequence is not convergent, i.e. the space under consideration is an incomplete normed linear space. Note that, even if we find one Cauchy sequence in this space which does not converge, it is sufficient to prove that the space is not complete.

Example 35

Let MATH i.e. set of rational numbers ($Q$) with scalar field also as the set of rational numbers ($Q$) and norm defined as MATH                                -------- (23) A vector in this space is a rational number. In this space, we can construct Cauchy sequences which do not converge to a rational numbers (or rather they converge to irrational numbers). For example, the well known Cauchy sequence MATH converges to $e$, which is an irrational number. Similarly, consider sequence MATH Starting from initial point $x^{(0)}=1,$ we can generate the sequence of rational numbers MATH which converges to $2+\sqrt{3}$ as MATHThus, limits of the above sequences is outside the space $X$ and the space is incomplete.