The Fredholm alternative for elliptic problems now says:
This is based on applying the Fredholm alternative for compact linear
operators K : H ->H to the K below.
Correction above: skip the last = 0. The condition is:
(h,v)=0 for all v with v = K* v
Remark 1. Every orthonormal sequence in a Hilbert space
converges weakly to zero.
Remark 2. Every infinite-dimensional separable Hilbert space
has a countable Hilbert basis and is thereby isometrically
ismomorphic to little (l)^2, the space of square integrable sequences
with the "Euclidean" norm. The above statement about weakly
convergent subsequences is easily proved directly in this space,
without any use of weak topologies or whatsoever.