4. Prove that the function that sends an ordered pair (p,q) of elements in P(R) to * p(x) q(x)e da defines an inner product on P(R). 0

3. For each of the following mappings, write out and for the given and, where.

For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.