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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 $V$?

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Is $W'$ a subset or a sub space? Every subspace of a vector space is a direct summand: take a basis for $W'$ and extend it to a basis for $V$. – Dylan Moreland Feb 23 '12 at 18:33
[Of course, if $W'$ is not a subspace then there is no hope. Look at the definitions.] – Dylan Moreland Feb 23 '12 at 18:39

$W'$ 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.

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$W$ 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 $W$.

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