It's a basis! Nobody cares!
Jan. 30th, 2006 02:36 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoyArgh!
So Mr. Sally gave us for homework - I'm pretty certain this was an open-ended thing he didn't expect us to get - if there could be any isometries of lpn(C) fixing 0 (for p≠2) other than compositions of conjugations of coordinates, multiplications of coordinates by units, and permutations of the coordinates. Well we know the linear isometries are just compositions of permutations and multiplications by units, so can we show it must be conjugations and something linear? Well, I managed to figure out that if we have V,W complex vector spaces, T:V→W bijective and R-linear, then T can be written as T_L+T_CL where T_L linear, T_CL conjugate-linear, and we can write V as the direct sum of V_L and V_CL, where T=T_L on V_L and T=T_CL on V_CL; so, taking a basis for these two subspaces, putting them together to get a basis for V, we find that T must be taking conjugates of some of the coordinates with respect to this basis (if we like, we can even make it a basis of norm 1 assuming we've got a norm), followed by something linear.
Now, I'd say this is pretty cool, but the problem is that this is ultimately useless as we have no assurance that this basis is, in fact, the standard one. In fact, there's an isometry of l22(C) which cannot be written as conjugations (wrt the standard basis, I mean :P ) followed by something linear. Of course, we're only interested in p≠2 right now, but the point is that we're basically reduced to showing it directly from properties of lpn(C) for p≠2, i.e., that we haven't actually made a bit of progress at all.
Conclusions:
I'd guess the only isometries of lpn(C) for p≠2 are just permutation, conjugation, multiplication by units, but I have no idea.
The linear isometries of l2n(C) are already more complicated than for p≠2, but the nonlinear ones just get *nasty*. (Unless perhaps *any* isometry of it considered as a real NLS would be an isometry of it considered as a complex NLS... that would be kind of nice and kind of disgusting. I should see if that's true.)
-Sniffnoy, very grateful for Mazur-Ulam
So Mr. Sally gave us for homework - I'm pretty certain this was an open-ended thing he didn't expect us to get - if there could be any isometries of lpn(C) fixing 0 (for p≠2) other than compositions of conjugations of coordinates, multiplications of coordinates by units, and permutations of the coordinates. Well we know the linear isometries are just compositions of permutations and multiplications by units, so can we show it must be conjugations and something linear? Well, I managed to figure out that if we have V,W complex vector spaces, T:V→W bijective and R-linear, then T can be written as T_L+T_CL where T_L linear, T_CL conjugate-linear, and we can write V as the direct sum of V_L and V_CL, where T=T_L on V_L and T=T_CL on V_CL; so, taking a basis for these two subspaces, putting them together to get a basis for V, we find that T must be taking conjugates of some of the coordinates with respect to this basis (if we like, we can even make it a basis of norm 1 assuming we've got a norm), followed by something linear.
Now, I'd say this is pretty cool, but the problem is that this is ultimately useless as we have no assurance that this basis is, in fact, the standard one. In fact, there's an isometry of l22(C) which cannot be written as conjugations (wrt the standard basis, I mean :P ) followed by something linear. Of course, we're only interested in p≠2 right now, but the point is that we're basically reduced to showing it directly from properties of lpn(C) for p≠2, i.e., that we haven't actually made a bit of progress at all.
Conclusions:
I'd guess the only isometries of lpn(C) for p≠2 are just permutation, conjugation, multiplication by units, but I have no idea.
The linear isometries of l2n(C) are already more complicated than for p≠2, but the nonlinear ones just get *nasty*. (Unless perhaps *any* isometry of it considered as a real NLS would be an isometry of it considered as a complex NLS... that would be kind of nice and kind of disgusting. I should see if that's true.)
-Sniffnoy, very grateful for Mazur-Ulam
