Direct sum of subspaces of a vector space.

Question

Let $V$ be a vector space and ,

$V = \quad \bigoplus_{i=1}^{n} w_{i}$

where $w_{i}$ are subspaces of $V$ and $w_{1} \bigcap w_{2} \bigcap ....w_{n} = \{0\}$ . My question is do we need pairwise disjointness as a sufficient condition.Is there any counter example for the same ? Is this true for Infinite dimensional vector spaces ?

Thanks in Advance.

Answer 1

As a wild guess, I'm guessing that you're asking, "If the $w_i$ have intersection 0, is $V$ the direct sum of them? Or do they need to pairwise intersect in 0?"

The answer is "You need pairwise intersection." Think of the $xy$, $yz$, and $xz$ subspaces of $\mathbb R^3$: the vector $(0, 1, 0)$ can be written as a sum of elements of the first and third, or as an element of the second, so $\mathbb R^3$ is not a direct sum of these subspaces, but the intersection of the subspaces is trivial.