Let T on ell2 have diagonal weights 1/n. Its bounded graph is closed in ell2
direct sum ell2, and projection to the second summand has image equal to the
range of T.

The truncations of y_n=1/n lie in the range and converge by the tail bound
sum_{n>m} 1/n^2 <= 1/m. The forced preimage is the all-ones sequence, whose
squared norm diverges, so the limit has no preimage in ell2. The exact
coordinates, partial norms, tail parameters, and proof obligations are in the
structured submission.
