though this disqualifies the set (pk) for membership in 52, they nevertheless an the space. They comprise a valid set of basis vectors and the projection of y function in 52 onto any member of the basis (pk) gives a finite result. If p is y function in 52, then (414)≤8 (4.46) me functions {k} may, through proper renormalization, be cast in a form which lows them to be members of 52. (See Problem 4.6.) ROBLEMS 4.6 Consider the functions k = eikx √a defined over the interval (-a/2, +a/2). (a) Show that these functions are all normalized to unity and maintain this normal- ization in the limit a →oo. (b) Show that these functions are an orthogonal set in the limit a → ∞. 4.7 State to which space each of the functions listed belong, 1 or 2. (a) f1 = (x-x-ax4 + ax³)/(x-2a) (b) f2 = (sin x)e-x2 (c) f3=√√In[x(x-a)+1] (d) f4 = sin 2nл [x(x-a)+1], n = 0, 1, 2,... (e) fs = elax (x²+a²)-1 (f) f6=x10e-x² (g) f1/sin kx 4.8 The function 9(x) = x(x - a)eikx is in 51. Calculate the coefficients of expansion, an, of this function, in the series representation (4.33), in terms of the constants a and k. Use the basis functions (4.15). 4.9 Two vectors and op in Hilbert space are orthogonal. Show that their lengths obey