Sections 1.1 and 1.2 See Theorem 1.1. Note that ||K f|| bounded by C||f|| is used to show the limit is a solution. k(x,t) was missing in the calculation above, see yellow correction below. Below: suppose we can get close to f with a linear combination of the phi's. The f_n made with those phi's is even closer, using Pythagoras. This definition says we can get close to any f with a linear combination of the phi's. Below we see that consequently the coefficients (f,phi_i) form an l_2 sequence with the same norm as f. Thus we have our map from H to l_2. For H separable apply the G-S procedure to the sequence in the definition of separability to obtain a complete orthogornal sequence