I have a Hilbert space H; given a closed subspace U of H let P(adsbygoogle = window.adsbygoogle || []).push({}); _{U}denote the orthogonal projection onto U. I also have a lattice L of closed subspaces of H, such that for all U and U' in L, P_{U}and P_{U'}commute. The problem is to find an orthonormal basis B of H, such that for every element b of B and every element U of L, b is an eigenvector of P_{U}(equivalently, b is in U or U^{⊥}).

The obvious thing to do is to apply Zorn's lemma to obtain a maximal orthonormal subset B of H satisfying the above condition, and this part works. For some reason or other, though, I'm having trouble showing that span B = H. If not, then letting W = span B, I need to find a normalized vector v in W^{⊥}such that for every U in L, v is an eigenvector of P_{U}; then B ∪ {v} contradicts the maximality of B. (The following may or may not be helpful: It suffices to consider the case where U contains W.)

The idea I have right now is this: Suppose I could find a one-dimensional subspace V of W^{⊥}such that P_{V}commutes with P_{U}for all U in L, and let v be a normalized vector in V. Then for every U in L, P_{V}P_{U}(v) = P_{U}P_{V}(v) = P_{U}(v), so P_{U}(v) is in V. Since V is 1-dimensional, v is an eigenvector of P_{U}(v), as desired.

The problem is that I have no idea how to choose V. I feel like this should be really easy, but for some reason I'm not seeing it.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Finding an orthonormal basis of a Hilbert space relative to a lattice of subspaces

**Physics Forums | Science Articles, Homework Help, Discussion**