If $V=V_1\oplus V_2\oplus\dots \oplus V_n$, then is it necessary that $V_i\cap V_j=\{0\}$?

Question

Say $$V=V_1\oplus V_2\oplus\dots \oplus V_n$$ where $V$ is a vector space and $V_1,V_2,\dots, V_n$ its subspaces.

Is it necessary that $$V_i\cap V_j=\{0\}$$ I think it is, for unique representation of a vector, but I am not sure.

Thanks

Answer 1

That any two distinct subspaces $V_i$ having a zero intersection is a necessary condition for the sum to be direct is immediate from the definition of a direct sum. If $v\neq 0$ were a vector in the $V_i\cap V_j$ then one could write $0=0+\cdots+0+v+0+\cdots+0-v+0+\cdots+0$ with the nonzero terms in positions $i,j$, showing that the expression for the zero vector as element of $V_1+\cdots+V_n$ is not unique.

Beware though that this is very far from being a sufficient condition if $n>2$.