1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 + |19|12

(b) Let X be the vector space of all sequences of complex numbers with only finitely many nonzero terms. Consider the following two norms: ||. ||1 and ||. ||2 on X: ||c||1 V -j=1 where x = (x;). Prove that ||. || 1 and || . ||2 are not equivalent.

1. Let V be an inner product space. Suppose f,g E V such that f - g Lg (a) Prove that 1l12= ||f- g|2 + ||9||2.

The set Fun(R) of all real-valued functions defined on R., together with the standard function addition and scalar multiplication, is a vector space. Select all the vectors that belong to Fun(R) A. (x² – 3z – 2)² OB. 0 Oc. VE OD. arctan(z) IE. OF. sin(z) OG. In(z)