What is a vector space?
A vector space is just a set V of vectors on which are defined two operations: addition and multiplication. Note: They don't actually have to be addition and multiplication, they just have to hold for these ten axioms, and addition and multiplication hold.
The sum of vectors u and v, denoted u + v is in V (closure under addition)
u + v = v + u
(u + v) + w = u + (v + w)
There is a zero vector 0 (doesn't actually have to be 0) in V such that u + 0 = u
For each u in V, there is a vector -u in V such that u + (-u) = 0
The scalar multiple of u by c, denoted by cu is in V. (closure under scalar multiplication)
c(u + v) = cu + cv
(c + d)u = cu + du
c(du) = cdu
1u = u
Basically, showing that it's closed under addition and scalar multiplication makes it a vector space given our normal + and *, because of the properties of real numbers. When the operations aren't orthodox, you must show all 10.
What is a subspace?
A subspace is a subset of a vector space that has three properties
It has the zero vector
It's closed under vector addition (Axiom 1 from before)
It's closed under scalar multiplication (Axiom 6 from before)
So, a subspace is a vector space. Conversely, every vector space is a subspace too! (of itself or larger spaces)