As per my understanding "for some" means a few elements of. It refers that the statement is true for at least one element but needn't be true for all.
In the book "An Introduction To Linear Algebra" by Arora, J. L on page 118 (question 3 of problem set 4.2) , "for some" is interpreted as "for all" since that is only when the statement is true.
Is this simply a case of confusing use of the phrase or this how "for some" is to be interpreted?
This could be phrased as "Define $T(u)$ to be the collection of all $w \in W$ for which there exists at least one $u \in U$ such that $w = T(u)$". The statement overall is "show that the image of a subspace under a linear map is a subspace", which is true.