Determine if each of the following sets with the indicated operations of addition and scalar multiplication are a vector space. If so, verify the axioms, and if not, indicate at least one axiom that fails. i) Let V be the set of all positive real numbers. Define the operations of addition and scalar multiplication Ⓒ by x + y = xy and x©y=x² where the operations on the right side of each equation is usual real number multiplication and exponen- tial, respectively. ii) Let V denote the vectors in R2 with the usual vector addition, but scalar multiplication is instead defined as I -- C = CI

Determine if the vector product <A,B> = a0b0 defines a valid inner product on P2(R) for any A=a0+a1x+a2x^2 and B=b0+b1X+b2x2. Explain your answer clearly.

If two Hermitian operators A and B commute, show that they can be simultaneously diagonalized.

Show that the set of all real polynomials with a degree n < 3 associated with the addition of polynomials and the multiplication of polynomials by a scalar form a vector space.

The set containing the elements specified also possesses its relevant operations of multiplication over reals and vector addition. Is the set a real vector space?