why in spectral theory , spectral subspaces are viewed as the span of the generalized eigenvectors (, roughly speaking the generalized eigenvectors as i have understood are kind of eigenvectors of an operator although they don't belong to the Hilbert space on which the operator act. ) I have been reading Quantum Theory for Mathematicians by Brian ,i will quote what he said hooping to make my question more clear .
note : my question dealt with proposition 5 in Brian's quote
Given a bounded (for now) self-adjoint operator A, we hope to associate with each Borel set E ⊂ σ(A) a closed subspace V E of H, where we think intuitively that V E is the closed span of the generalized eigenvectors for A with eigenvalues in E. [We could do this more generally for any E ⊂ R, but we do not expect any contribution from R\σ(A).] We would expect the collection of these subspaces to have the following properties. 1. V σ(A) = H and V ∅ = {0}. 2. If E and F are disjoint, then V E ⊥ V F . 3. For any E and F , V E∩F = V E ∩ V F . 4. If E 1 , E 2 , . . . are disjoint and E = ∪ j E j , then V E = V E j . j 5. For any E, V E is invariant under A. 6. If E ⊂ [λ 0 − ε, λ 0 + ε] and ψ ∈ V E , then (A − λ 0 I)ψ ≤ ε ψ . The condition V σ(A) = H captures the idea that our generalized eigenvec- tors should span H, while Property 2 captures the idea that our generalized eigenvectors should have some sort of orthogonality for distinct eigenval- ues, even if they are not actually in the Hilbert space. In Property 4, there may be infinitely many of the E j ’s, in which case, the direct sum is in the Hilbert space sense (Definition A.45). Properties 5 and 6 capture the idea that V E is made up of generalized eigenvectors for A with eigenvalues in E.