Guided Proof Let W be a subspace of the innerproduct space

QUESTION:

Guided Proof Let W be a subspace of the innerproduct space V. Prove that the setW = {v V: v, w = 0 for all w W}is a subspace of V. Getting Started: To prove that W is a subspace ofV, you must show that W is nonempty and that theclosure conditions for a subspace hold (Theorem 4.5). (i) Find a vector in W to conclude that it is nonempty. (ii) To show the closure of W under addition, youneed to show that v1 + v2, w = 0 for all w Wand for any v1, v2 W. Use the properties ofinner products and the fact that v1, w and v2, ware both zero to show this.(iii) To show closure under multiplication by a scalar,proceed as in part (ii). Use the properties of innerproducts and the condition of belonging to W.