I'm reading "Introduction to Differential Topology" by Brocker and Janich.
What other sum of vector spaces is there? I can see if we are trying to sum two identical spaces, it doesn't make much sense to construct $\mathcal{V} \oplus \mathcal{V}$, a set of ordered pairs. Instead we would want to just say $\mathcal{V} + \mathcal{V} = \mathcal{V}$. Is this what the author is referring to? If so, what name do we give the $(+)$ operator in the preceding sentence? $(\oplus)$ is called the "direct sum," for example.
If we are given a vector space $X$ and two vector subspaces $V,W \subset X$, then $V+W = \{v+w \, | \, v \in V, w \in W\}$ is also a subspace. This kind of sum is what is used in studying transversality, where $X$ might be for example the tangent space of a differential manifold at some point, and $V,W$ might be tangent spaces of submanifolds passing through the same point. This sum will be a direct sum if and only if $V \cap W$ is the trivial subspace.