Proving the Existence of Direct Sums in Linear Algebra

[omoplata]

In 'Linear Algebra Done Right' by Sheldon Axler, a direct sum is defined the following way,

We say that $V$ is the direct sum of subspaces $U_1, \dotsc ,U_m$ written $V = U_1 \oplus \dotsc \oplus U_m$, if each element of $V$ can be written uniquely as a sum $u_1 + \dotsc + u_m$, where each $u_j \in U_j$.

Suppose $V = U \oplus W$. Is there any way I can prove that for all $u \in U$ there exists $v \in V$ and $w \in W$ such that $v = u + w$?

If that can be done, then I can solve a problem given later in the book.

 

[micromass]

Take $w=0$ and $v=u$.

 

[omoplata]

LOL, OF COURSE!

Thanks!