Suppose we assign in $V$ (vector space) a bilinear symmetrical form $b$. Let $S$ be a subspace of $V$, $S^{\bot} := \{ w \in V \mid b(v,w) = 0\ \forall w \in S \}$
Two subspaces $U$ and $W$ are said to be orthogonal if $U \subset W^{\bot}$ and this is equivalent to $W \subset U^{\bot}$.
Could someone spell the proof of this equivalence out for me I can't seem to see it.
Suppose that $U \subset W^\perp$.
If $w \in W$ and $u \in U$ then $b(w,u) = 0$ because $u \in W^\perp$. Thus $b(w,u) = 0$ for all $u \in U$, so that $w \in U^\perp$. It follows that $W \subset U^\perp$.