I have a question in linear algebra which I have been brain storming for days - and I thought I will resort to help.

Suppose that we have two finite dimensional Hilbert spaces H1 and H2. I need to construct a basis B for

H = H1 $\bigoplus$ H2

such that for any two vectors D1,D2 in H1, vectors E1,E2 in H2 and coordinate representation V1 of (D1,E1) on B and coordinate representation V2 of (D2,E2) on B, if V1 and V2 are "close" then D1 should be "close" to E1 andD2 and E1 should be "close" to E2.

I have a question in linear algebra which I have been brain storming for days - and I thought I will resort to help.

Suppose that we have two finite dimensional Hilbert spaces H1 and H2. I need to construct a basis B for

H = H1 $\bigoplus$ H2

such that for any two vectors D1,D2 in H1, vectors E1,E2 in H2 and coordinate representation V1 of (D1,E1) on B and coordinate representation V2 of (D2,E2) on B, if V1 and V2 are "close" then D1 should be "close" to E1 and D2 should be "close" to E2.

In other words, B should "preserve" the norm if the vectors on B are projected on the respective subspaces.

I think such problems would have been encountered in physics (QM) where one needs a frame of reference that preserves the norm (I am really vague here, sorry). So physics intuitions are highly welcome...

Also, any help in further refining the problem will be greatly appreciated.

Thanks