If U is a vector space (doesn't need to be a subspace of a larger vector space V) then U + [any subspace of U] = U. This is easily derived from considering the basis vectors of U and any of it's possible subspaces.
That depends on how you define addition of vector spaces.
Personally I would use direct sum, when there is a well defined identity, the zero vector space.
You cannot add a subspace of U to U whilst considering it as a subspace. You can add a vector space isomorphic to a subspace of U (which is just any subspace of smaller dimension in fact) and get something strictly different from U, barring adding the zero vector space).
Vector addition is defined as the set of all possible sums of vectors from the two vector spaces.
There's something I don't understand about direct sums. It's not a binary operator or anything, so V direct sum W could possibly produce nothing at all. It's like saying 'A disjoint union B'. All we can say is either the statement is true or false. So how does one 'use' direct sums?
The addition you define is that which operates on subspaces of some given vector space. It isn't 'vector addition', it is addition of subspaces, perhaps even 'addition of vector subspaces'.
Anyway, I think you have a misconception about the (entirely different) subject of direct sum.
Why is direct sum not a binary operator on the category of vector spaces (or any other abelian category)? I'd be interested to hear why you think this, since it certainly is a binary operator, actually it can be an n-ary operator, or possibly even an A-operator for any integer n, or cardinal A.
Direct sum must take place in some ambient category, and it is never the case that [itex]V\oplus W[/itex] is zero when V and W are non-zero in the category of vector spaces.
The direct sum of A and B is the smallest vector space containing both A and B as subspaces intersecting in 0 only.
It is the vector space analogue of disjoint union (coproduct), and I am perplexed by what "All we can say is either the statement is true or false" has to do with anything.
it is two copies of the object, each having the canonical injection from one copy of the set. that is the disjoint part of disjoint union.
we are after all only dealing with things upto isomorphism. thus it is just a set with twice as many elements and a pair of maps.
How do you take the direct sum of a vector space with itself?