TITLE: Let U, W be subspaces of a vector space V. Suppose U ⊆ W. Prove or disprove: U + W = W
QUESTION [5 upvotes]: So, I know that W + W = W. And it makes sense that there is no counterargument that the claim isn't true.
So, here is my attempt:
Claim: $U + W = W$
Proof:
$$W \subset U + W$$
Let $w \in W$. Then $w = w + 0$, which is a sum of an element from $W$ and an element of $U$. Thus, $w \in W + U$, and hence $W \subset U + W$.
$$U + W \subset W$$
By definition, $U + W = \{u + w ~:~ u \in U \text{ and } w \in W\}$. That is, every element $v \in U + W$ is of the form $v = u + w$, with $u \in U$ and $w \in W$. And since $U \subseteq W$, it follows that $u \in U$ implies $u \in W$. Thus, every element of $U + W$ is a sum of two elements of $W$, and hence $u + w \in W$ since $W$ is a subspace. Thus, $U + W \subset W$.

REPLY [2 votes]: Your proof is correct. I would just mention that $U+W$ is the smallest subspace containing $U$ and $W$ so with the hypothesis $U\subset W$ it's clear that $U+W=W$.