[Picture] [Picture] [Picture] The Fredholm alternative for elliptic problems now says: [Picture] This is based on applying the Fredholm alternative for compact linear operators K : H ->H to the K below. [Picture] [Picture] Correction above: skip the last = 0. The condition is: (h,v)=0 for all v with v = K* v [Picture] 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. [Picture] [Picture] [Picture] [Picture] [Picture] [Picture]