Let $X$ be a norm linear space. If $X$ is banach space then subspace $Y$ is closed iff $Y$ is complete.
But if $X$ is not banach space then $Y$ is closed need not imply $Y$ is complete.
Can u give me such an example?
I took a non-banach space $C[0, 1]$ with integration norm. But I am unable to find such an example here.
Closed subspaces don't have to be complete. You could take any incomplete metric space and then the space is closed with respect to itself. However, for a subspace, take the rationals in any closed interval, which will be closed in $\mathbb{Q}$ but not complete.
Take the normed space $P$ of polynomials in $[0,1]$ with the supremum norm. This space is not complete, as we can find a squence that is Cauchy and converges to $e^x$ in $C[0,1] \setminus P$. Namely,
$$ p_n := 1+ x + \dots + \frac{x^n}{n!} $$
converges to $e^x$ in $C[0,1]$ with the supremum norm, so the sequence in $P$ must be Cauchy but it doesn't converge there. We also have that
$$ p_n \in \{p \in P : p(0) \geq 1 \} =: F, $$
so in particular the closed set $F \subset P$ is not complete.
