is the set of real numbers.
is the set of n-tuples of real numbers.
I would assume that the notation means that you're looking at the vector space of n-tuples of real numbers over the field of reals where the operation is normal multiplication.
Now, the null vector is in the subset because
If are in the subset, then
So is in the subset.
Scalar multiplication is similar.
Yes, it's a subspace.