Vector subspace equality proof/disproof. Given $R,S,T$ are subspaces of vector space $V$, and $R+S=R+T$, does it follow $S=T$?
Please don't give a full proof, but some general help would be much appreciated. I get the basic idea that to show $S=T$ would be to show them to be subsets of one another. Not sure how to do this in a concrete way though.

Another kind of example in infinite dimension:
Take $V=\mathbb R[X]$ and


*

*$R=\mathbb R[X]_{\leq 6}$

*$S=\mathbb R[X]_{\leq 3}$

*$T=\mathbb R[X]_{\leq 2}$.
Then $R+S=R+T$ but $S\ne T$.
