Orthogonal complement by example

Question

I know what orthogonal complement is. Let W be subset of inner product space V whose orthogonal complement we are considering.

Now, orthogonal complement of W is equal to orthogonal complement of orthogonal complement of orthogonal complement, i.e. W(perpendicular)=W(3×perpendicular)

I know how to prove it using definition. But I want to know how this is possible .help me to understand .

Also, why W is subset of W(2×perpendicular)?Not able to understand this.

Thanks!

Answer 1

Let W={(1,0,0),(0,1,0)} be subset of inner product space.

Think geometrically the situation. W (perpendicular )is thus z- direction. W (2×perpendicular) is xy plane . As we can see W is subset of W (2×perpendicular) here. Again ,W (3×perpendicular ) is z- direction which is equal to W (perpendicular).