Homework 4 Jeff

be an orthonormal basis of H consisting of eigenvectors of A where {ek : k ∈ N} is a set of eigenvectors of A corresponding to nonzero eigenvalues of A and {fm : m ∈ I} is a set of eigenvectors of A corresponding to the zero eigenvalue (we know that we may assume that I is a countable set by exercise 6.10). We make the assumption that the elements of {ek : k ∈ N} are ordered so that the first element(s) of this set (ordered by subscript) are eigenvectors corresponding to the eigenvalue λ1, the next element(s) are eigenvectors corresponding to the eigenvalue λ2, and so forth.