Let x = (x1, ... ,xn) and y = (y1,...,yn) ∈ Rn be fixed. Recall that Rn is a vector space, and that it is an inner product space when equipped with ⟨x,y⟩ = Σ nj=1 xnyn, where Σ is the summation notation.  This problem walks you through in proving the Cauchy-Schwartz inequality for this inner product. (a) For z ∈ R, Verify the following identity:P(z) = (x1z + y1)2 + (x2z + y2)2 + ··· + (xnz + yn)2          = (Σnj=1 x2j )z2 + 2 (Σnj=1? xjyj)z + Σnj=1 y2j. (b)  Explain why this polynomial P(z) is always nonnegative.

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The conjugate space H" is also a Hilbert space with respect to the inner product defined by (fx. fy) = (y,x).

2. Check that the following are not vector spaces by proving that at least one axiom in the definition of a vector space is false. (a) The quadruple (R, R", H, O2). Here, H denotes the usual operation of addition multiplication among vectors, while O2 denotes the nonstandard operation of scalar multiplication given by c(x1, x2, . .. , Xn) = (cx1,0, 0, . . .,0). (b) The quadruple (R, V, H, D), where V denotes in this case the subset of Rmxn whose elements have the entry in row 1 and column 1 equal to some r E R, r + 0. Here, B and O denote the usual operations of addition and scalar multiplication among matrices.