The space is complete iff the subspace and the quotient space is complete.
95887
How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
67191
When is the union of a family of subspaces of a vector space also a subspace?
10907
Necessity of completeness of the inner product space in Riesz representation theorem
134065
A question about an $n$-dimensional subspace of $\mathbb{F}^{S}$.
41047
Subspace Topology of a subset
106408
Prove that if W is a subspace of a finite dimensional vector space V, then dim(W) ≤ dim(V).
60684
Determine whether the set $W$ is a subspace in $\mathbb{R}^{3}$
132487
Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$
104578
Suppose that $V$ is a vector space, and $W$ is a subspace of $V$. If $V$ is finite dimensional, then prove $W$ too must be finite dimensional.
100362
Codomain of function complete implies function space complete
128894
About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$
114524
Proof Verification: the orthogonal complement of the column space is the left nullspace
6523
Prove that the space has a countable dense subset
74312
If Y is complete then B(X,Y) is complete
154324
preservation of completeness under homeomorphism
119105
If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
123665
A question in Subspaces in linear algebra