Let $V$ be a vector space and $W \subset V$ be a direct summand of $V$. If W' \subset W then under what conditions is W' a direct summand of $V$?
Conditions for a subset of a direct summand of a vector space to be a direct summand
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1[Of course, if $W'$ is not a subspace then there is no hope. Look at the definitions.] – 2012-02-23
2 Answers
W' is a direct summand of $V$ if and only if W' is a subspace of $V$, as others have already pointed out. In general, it doesn't make sense to talk about direct sums unless the summands are subspaces. For completeness sake, here's a short proof that W' is a direct summand of $V$ whenever W' is a subspace of $V$:
Let $\{w_i\}$ be a basis for W'. We can extend this basis to a basis $\{w_i\}\cup\{v_j\}$, either by adding in linearly-independent vectors one-by-one (in the case of finite dimensional spaces), or by Zorn's Lemma (the proof is the same as the proof that every vector space has a basis, only we consider linearly independent sets that contain $\{w_i\}$ instead of all linearly independent sets).
W' is the span of $\{w_i\}$, and we can let $W_1$ be the span of the remaining basis vectors. Then V=W'\oplus V.
W' and $W_1$ are called complementary subspaces. Note that our choice of $W_1$ depended on our choice of basis, so that in general W' will have many different complements.
$W$ is direct summand of $V$ that is we have some vector subspace of $V$ call it $W_1$ such that $V= W\oplus W_1$ Now W'\subset W. For being a direct summand of $V$, W' should be a subspace of $W$. That is W' should be a subspace of $V$ contained in $W$.