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## Main Question or Discussion Point

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

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.

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.