There is no definition of the product of two elements in ?!
You put a product with a scalar and an element of , but not between elements of . Are you sure no information about the problem is missing?
Anyway, I didn't check all but for the 1) so is not a vector space.
(Detail: while ).
And about the 2), it seems to be a vector space but as you didn't define the product of two elements in V, we cannot affirm it.