Spectral theorem for self-adjoint continuous operators on Hilbert spaces

Let T be a continuous linear map V → V for a (separable) Hilbert space V . Its spectrum σ(T ) is a compact subset of R, so is certainly contained in some finite interval [a, b]. As usual, for a self-adjoint continuous operator S on V , write S ≥ 0 when 〈Sv, v〉 ≥ 0 for all v ∈ V . For self-adjoint S, T , write S ≤ T when T − S ≥ 0. At the outset, with a ≤ −|T |op and b ≥ |T |op, we have, 〈a · v, v〉 ≤ 〈Tv, v〉 ≤ 〈b · v, v〉. That is, a ≤ T ≤ b, where the scalars refer to scalar operators on V . As a corollary of part one of the spectral theorem, we will see that, in fact, infλ∈σ(T ) λ ≤ T ≤ supλ∈σ(T ), but it seems difficult (and un-necessary) to prove this inequality directly.