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.

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

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