Consider a set S not necessarily a subspace of an inner prod

Consider a set S (not necessarily a subspace) of an inner product space V. Define S = {u V| = 0 for all w S}. Prove that S is a subspace of V.

Solution

We will denote S perpendicular by S\'

1. Let, u,v belong to S\'

So,

<u,w>=<v,w> for all w in S

<u+v,w>=<u,w>+<v,w>=0 for all w in S

Hence, u+v is in S\'

2. Let, u in S\' and c be scalar

<cu,w> =c<u,w>=c*0 for any w in W

Hence, cu is in S\'

Hence, S\' is subspace of V