For a subspace W of an inner product space V define its ort

For a subspace W of an inner product space V , define its orthogonal complement W by

W ={ v V | v,w = 0 for all w W }. Show that W is a subspace of V .

Solution

Denote orthogonal complement by W\'

1. Check for closure under addition

LEt, x,y be in W\'

So for any v in W

<x+y,v>=<x,v>+<y,v>=0

So closed under addition

2. Check for closure under multiplication

Take x as in 1. and c any scalar

<cx,v>=c<x,v>=c*0=0

So closed under scalar multiplication

So W\' is a subspace of V