Show us your proof for part (b). There's probably a step in there that doesn't necessarily hold for infinite dimension subspaces.(a) and (b) are fairly traditional, but I have trouble understanding the phrasing of (c). What makes the infinite dimensionality in (c) different from (a) and (b)?
Yes B is not that hard for finite dimensional spaces - as a hint just look into the spectral theorem. I can write the proof in a few lines using bra-ket notation. Extending that to infinite dimensional spaces is much more difficult (it requires the so called Nuclear Spectral theorem - also called the Gelfand-Maurin theorem) which may be the purpose of the question - infinite dimensional spaces are problematical.Show us your proof for part (b). There's probably a step in there that doesn't necessarily hold for infinite dimension subspaces.