  Two more definitions before we can say what Hilbert spaces are, and then we can turn to quantum mechanics. \(\ket{B}\) is an  eigenvector of \(O\) with eigenvalue \(a\) if, and only if, \(O \ket{B} = a \ket{B}\). Different operators can have different eigenvectors, but the eigenvector/operator relation depends only on the operator and vectors in question, and not on the particular basis in which they are expressed; the eigenvector/operator relation is, that is to say, invariant under change of basis. A Hermitean operator is an operator which has the property that there is an orthonormal basis consisting of its eigenvectors and those eigenvalues are all real. 