A subspace Y of a Banach space X is a subspace of X considered as a normed space. Hence we don’t require Y to be complete. is this true? I cannot come out with an example to this
It's true. Take, for instance the set of all sequences in $c$ that are eventually 0 (that is, of the form $(x_1,x_2,\ldots, x_n,0,0,\ldots)$ for some $n$. Here, $c$ is the space of all bounded sequences with limit 0 endowed with the $\sup$ norm.
More generally: It is known that an infinite dimensional Banach space cannot have a countable Hamel Basis. From this, it follows that the set of finite linear combinations of elements from a countably infinite, linearly independent subset of $X$ cannot be closed (closed subspaces of Banach spaces are Banach spaces).