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.