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