$S$ is a subspace of $V$, then does $S$ perp contain $V$ perp? My question is very simple.
If $S$ is a subspace of the vector space $V$, would that make $V^{\perp}$ contained by $S^\perp$? 
I am asked to prove this theorem, but I couldn't move a pencil :(

Assume $S\subset V\subset X$, where $X$ is a vector space and $\subset$ is understood to mean vector subspace. Now, apply the definitions:
$$S^\perp=\{x\in X\colon \left<x,s\right>=0 \text{ for all } s\in S \}$$
$$V^\perp=\{x\in X\colon \left<x,v\right>=0 \text{ for all } v\in V \}$$
Hint: suppose $x\in V^\perp$. What can you say about $\left<x,s\right>$ where $s\in S$?
Remark: I have assumed $V\subset X$ since if $V$ is the universe, then $V^\perp=\{0\}$ and the claim is trivial.
