Is $\ell_0$ space a Banach space? $\ell_0$ space contains all infinite sequence which only has finite nonzero terms. Could anyone tell me whether it is a Banach space? Is it possible for us to find a Cauchy convergent series which is not convergent in this space?

Your vector space has Hamel dimension $\aleph_0$.  Every infinite-dimensional Banach space has Hamel dimension ${}\ge 2^{\aleph_0}$.  So there is no norm on your vector space making it a Banach space.  See Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.
