Is subspace of a vector space a quotient space?

Question

Reasoning:

1. Given a vector space V and a subspace W of V.
2. "x≡y (mod W) if x-y∈W" defines equivalence relation.
3. equivalence relation partitions V into equivalence classes.
4. equivalence classes is called cosets of W in V.
5. Define the quotient space V/W, whose elements are cosets of W in V.

My question:

If x & y ∈ W, x-y∈W. So it seems W⊆quotient space since W is a coset of itself. True? If yes, why V/W?

Answer 1

There is an element of the quotient space that has the same members as $W$, so $W$ is an element of $V/W$, but no, $W$ is not a subset of the quotient space. A subset of the quotient space would be a collection whose members are equivalence classes.

You are right when you say that $W$ is a coset of itself though, and that is why $W$ is an element of $V/W$.