If B is a subset of inner product space V, show orthogonal complement of B is a subspace of V

Question

Let V be a inner product space and B be a subset (not subspace) of V.

Is then B┴ a subspace of V ?

There is a guy here that allegedly proves so, but I can't quite follow. Any heads-up or a counter-example?

Answer 1

This is indeed true and not very difficult to prove. Just take the definition of $B^\perp$: $$v \in B^\perp \Longleftrightarrow \forall\, b \in B: \langle v,b \rangle = 0.$$ Now ask yourself, if $v,w \in B^\perp$ and $\lambda \in \mathbb{C}$, do we have $\lambda v + w \in B^\perp$?