Countable orthonormal basis of product of separable Hilbert spaces

Asked Apr 26 '13 at 19:12

Active Apr 26 '13 at 19:12

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If I have 2 separable Hilbert spaces $X$ and $Y$ which have (different) orthonormal bases $x_i$ and $y_i$, then clearly $x_i \times y_j$ is a basis for $X \times Y$ (which is also a separable space).

But it is not orthonormal. Do I have to use a Gram-Schmidt thing or is there a nice way of renorming or something to make it orthonormal? I mean, is there a nice formula?

I gave the product space the inner product $(\cdot,\cdot)_X + (\cdot,\cdot)_Y$.

asked Apr 26 '13 at 19:12

maximumtag

${x_i\times y_j}$ is not a basis. Try ${x_i\times 0}\cup{0\times y_j}$ – Diego Apr 26 '13 at 19:17
By saying "product", do you mean direct sum or tensor product? – 23rd Apr 26 '13 at 19:17
@Landscape I mean the direct sum. – maximumtag Apr 26 '13 at 19:22
@Diego Well it's dense, I believe.. – maximumtag Apr 26 '13 at 19:26
Then following the notations in the wikipedia page, an orthonormal basis of $X\oplus Y$ should be ${(x_i,0),(0,y_j):\forall i,j}$, which is essentially what @Diego means. – 23rd Apr 26 '13 at 19:27
@Landscape Unless I'm very confused, $((x_i,0),(0,y_j))_{X \times Y} = 0$ even when $i=j$. How can this be? – maximumtag Apr 26 '13 at 19:30
Your equation means the two vectors on the left hand side are othorgonal, which is compatible with the definition of orthonormal basis. What confuses you? – 23rd Apr 26 '13 at 19:34
I suggest you should recall the definition of orthonormal basis. – 23rd Apr 26 '13 at 19:36
Even when $i = j$, you're still dotting two different, orthogonal vectors. However, if you do $\langle (x_i, 0), (x_i, 0) \rangle$ then you get something nonzero. – Christopher A. Wong Apr 26 '13 at 19:39
Oh yes, that's right, thanks, got confused @ChristopherA.Wong – maximumtag Apr 26 '13 at 20:38
@Landscape Thanks I got it. – maximumtag Apr 26 '13 at 20:38
You are welcome. You may post an answer by yourself. – 23rd Apr 26 '13 at 20:52

Countable orthonormal basis of product of separable Hilbert spaces

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