Let $\mathcal{H}$ be a (for simplicity seperable) Hilbert space and $\mathcal{B}(\mathcal{H})$ denote the space of bounded linear operators on $\mathcal{H}$. Since characteristic functions are total in $L^\infty$, we know by the Borel functional calculus that the orthogonal projections are total in $\mathcal{B}(\mathcal{H})$.
However, let $\mathbb{1}$ denote the identity on $\mathcal{H}$ and let $(e_n)$ be an orthonormal basis of $\mathcal{H}$, then $\mathbb{1}=\sum_n \langle \cdot,e_n \rangle e_n$ holds in SOT but not in norm (as it is not a Cauchy sequence).
In my opinion it is really surprising that we have totality of orthogonal projetions in $\mathcal{B}(\mathcal{H})$ but even for the simplest operator the most natural approximation doesn't work.
Therefore my question is: Why is this theorem heuristically true (beyond the class of compact operators)?
Best, Sebastian
edit: replaced dense by total (I often use those terms synonymously in TVS despite it is formally not correct)
