In 'Linear Algebra Done Right' by Sheldon Axler, a direct sum is defined the following way,(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Direct sums of vector spaces

**Physics Forums | Science Articles, Homework Help, Discussion**