Solve it as soon as poss

5.33. Show that the constant K in Example 29 can be improved upon, in that K = max(1, *1)- 5.34. If a : X x X - R is a continuous bilinear form on an inner product space X, show that lim a(un, Un) = a(u, v) 1-+00 if un - u and Un -+ v. 5.35. Let / : H; (0, 1) - R and a : /} (0, 1) x H} (0, 1) - R be defined by (4,v) = / (-1 - 4xju de, a(u,v) = / (x + 1ju'v dx, where H(0, 1) = (v E L'(0, 1) : V' E L' (0, 1), v(0) = v(1) = 0); this is a Hilbert space (see Chapter 7) with the inner product (1, v)= = / (u + u'v) de = (u, v): + (u', v'). Show that f is continuous, that a is continuous and Hy- elliptic, and verify that the unique element u satisfying a(u, v) = (6, v) for all v e H; (0, 1) is u(x) = 1'-r. |Hint: it may be necessary to use integration by parts. You may assume that a constant (' > 0 exists such that |lulz: s 5.36. Let a : H x H - R be a continuous, H-elliptic bilinear form, and define the bilinear form a : H x H - R by a(u, v) = a(u, v) + (u, ku)ra