## Constructing basis on the space of direct sums. [closed]

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 D2 and E1 should be "close" to E2.

In other words, B should "preserve" the closeness 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 closeness (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

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@Kasthuri: This question will probably be closed; see mathoverflow.net/faq for more information. BTW, I don't see why the union of a basis of $H_1$ and a basis of $H_2$ should not work. – Martin Brandenburg May 27 2012 at 19:39
I don't know why it needs to be closed - I saw the faq and it will be great if you can provide more info. Basis of H1 union basis of H2 will simply not work - think about it... – Kasthuri May 27 2012 at 21:30
To help you more - consider this simple example. Two vectors v1 = [1,-1] and v2 = [0,1] - which is in the direct sum of R and R. The distance between them is 1. However, the distance between -1 and 1 is 2. So, the projection could have larger distance than in the combined space... So this is not a trivial problem... – Kasthuri May 27 2012 at 21:57
@Kasthuri: I don't see any reasonable metric on the direct sum of R and R that would make the distance between [1,-1] and [0,1] equal to 1. Certainly the usual metric for a direct sum of Hilbert spaces doesn't do that. – Andreas Blass May 27 2012 at 23:28