Let 1 and 2 be inner0 products on a vector space V Define 3

Let (?,?)_1 and (?,?)_2 be inner0 products on a vector space V. Define (?,?)_3 via (u, v)_3 = (u, v)_1 + (u, v)_2. Also define (?, ?)_4 via (u, v)_4 = ((u, v)_1)((u, v)_2). Determine if (?, ?)_3 and (?,?)_4 will always be linear products on V.

Solution

<.,.>3 is an inner product because aadition of two inner product is an inner product but <.,.>4 is not an inner product because multiplication of two inner product is not an inner product.