I am reading Applied linear algebra: the decoupling principle by Lorenzo Adlai Sadun (btw very recommendable!)
On page 30 about direct sums on vector spaces it says:
Let $V$ be the space of continuous functions on a domain $U \subset \mathbb R^3$. Then $V \oplus V \oplus V$ is the space of continuous $\mathbb R^3$-valued functions on $U$.
What I don't understand is how this threefold direct sum of $V$ can lead to a statement about the codomain ("-valued" functions). But I just guess that I don't get it altogether - please enlighten me. Thank you!
I think what confuses me most is that we start with $\mathbb R^3$ and end up with it. It would come more natural if we started with $\mathbb R$ and after taking the direct sum three times would end in $\mathbb R^3$.