Preferred way to write elements of the direct sum of vector spaces Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is
their direct sum. Reading through the literature I found essentially two
ways of writing elements of $V\oplus W$.
1.) We have the 'product like description' $(v,w)\in V\oplus W$
2.) We have $v+w\in V\oplus W$
Is there a correct or preferred way to write the elements?
I mean the direct sum is the coproduct in the category of vector spaces
not the product, so the first one is eventually misleading 
(although the product $V\times W$ is the same here, this is only for finite many factors/terms) 
On the other side, the second one can eventually be misleading. For
example, consider $a+b+c+d \in V\oplus W$. What does $+$ mean here? Is it the
$+$ from $V$ or from $W$ or from the direct sum? Is $b \in V$ or $b\in W$ ? You can't say ....
 A: The first is not misleading, the notation $(v, w)$ does not imply product vs. coproduct, it's just notation for a tuple.  If we had infinitely many summands (so that product $\neq$ coproduct) then writing $(v_i)$ just indicates that you have a tuple indexed by $i$.  It does not tell you whether that tuple has finite support or not and when it does have finite support this doesn't imply that you're considering the coproduct, elements of the product can have finite support as well (it's just not a requirement for the product).
The notation $(v, w)$ just indicates tuples.  As with most notations you have to either say explicitly or make sure it's clear from context in what domain you consider that element to be living.
The notation $v + w$ is also just as good though you're correct that you have to be extra careful to say where things live because the notation no longer indicates that $v$ is from the left factor and $w$ is from the right.
So use whichever you like, there is no correct choice or preference.  The thing to remember about all notation is that no matter how clear you think it is, it's never perfect.  You will always need to include prose to explain your mathematics.
