TITLE: Prove that $V=U \bigoplus W \approx U \times W$
QUESTION [5 upvotes]: Let $V=U \bigoplus W$, $V \approx U \times W$.  Note that $U,W$, are finite dimensional subspaces of the vector space V, and also that $U \bigoplus W$ means $V=U+W$ and $U \cap W = \{0\}$
I'm really not sure how to go about this, because it doesn't seem to be true to me.  But after some research, it does seem to be true. Thanks in advance.

REPLY [4 votes]: By definition
$$
U \oplus V = \left\{ u + v \ \vert \ u\in U, v \in V \right\} \ ,
$$
plus the fact that $U\cap V = \left\{ 0\right\}$.
Also by definition
$$
U\times V = \left\{ (u,v) \ \vert \ u\in U, v \in V \right\} \ .
$$
Can we conclude anything from that in order to define isomorphisms
$$
f: U\oplus V \longrightarrow U\times V \qquad \text{and} \qquad g: U\times V \longrightarrow U\oplus V \quad \text{?}
$$