1. Let (X, x) and (Y, y) be inner product spaces over the same field F. Prove that: (a) The map : (X x Y)2 F, defined by < (x, y), (x', y') >= < x,x' >x + < y;Y' >y is an F inner product on X x Y. (b) < u+u', x+r' >x - < u-u', x a' >x= 2 < u, x' >x +2 < u', x >x for all x, x', u, u' e X. () 4 < , 2' >- < + u',+>x 1. Let (X, x) and (Y, y) be inner product spaces over the same field F. Prove that: (a) The map : (X x Y)2 F, defined by < (x, y), (x', y') >= < x,x' >x + < y;Y' >y is an F inner product on X x Y. (b) < u+u', x+r' >x - < u-u', x a' >x= 2 < u, x' >x +2 < u', x >x for all x, x', u, u' e X. () 4 < , 2' >- < + u',+>x