Are union and sum the same?

I know what [itex]U \cup W[/itex] is, but how is [itex]U + W[/itex] defined?

 

There many different meanings for mathematical notations, depending on the context. The page you gave is not talking about "+" in the context of sets. (In the context of sets, some books use [itex] A + B [/itex] to denote [itex] (A \cup B) - (A \cap B) [/itex].)

The page is talking about vector spaces. In that contex, I think [itex] U + W [/itex] means the vector space consisting of all vectors [itex] h [/itex] that can be expressed as [itex] h = u + w [/itex] where [itex] u \in U [/itex] and [itex] w \in W [/itex].

However, if you want to be sure of the meaning of "+" in a particular book, you must see what that book says it means. There is no "universal" meaning for it.

 

for vector spaces, particularly when U,W are subspaces of a vector space V,

U+W is a subspace of V, U∪W usually is not.