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 Post subject: Odd basic problemPosted: Wed, 25 Apr 2012 13:54:42 UTC
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A friend of mine posed the question if I have a basis of and a dual basis which has the usual pairing rule and for every i, can we conclude that for every ? Note the assumption is only that the original basis is normal, nothing about orthogonality is assumed, otherwise this is trivial. I personally feel like it should just come from the transpose isomorphism, but he feels there might be something more subtle. If there's a reference or a really solid proof that actually has the maximization via the analysis definition taking the sup over unit vectors, that would probably be most convincing. Thanks, and any help appreciated.

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 Post subject: Re: Odd basic problemPosted: Wed, 25 Apr 2012 14:23:00 UTC
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A friend of mine posed the question if I have a basis of and a dual basis which has the usual pairing rule and for every i, can we conclude that for every ? Note the assumption is only that the original basis is normal, nothing about orthogonality is assumed, otherwise this is trivial. I personally feel like it should just come from the transpose isomorphism, but he feels there might be something more subtle. If there's a reference or a really solid proof that actually has the maximization via the analysis definition taking the sup over unit vectors, that would probably be most convincing. Thanks, and any help appreciated.

No.

Take with , , , where is tiny. Then obviously (lifting the indices to show it is the dual), but has norm .

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 Post subject: Re: Odd basic problemPosted: Wed, 25 Apr 2012 14:33:30 UTC
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outermeasure wrote:
A friend of mine posed the question if I have a basis of and a dual basis which has the usual pairing rule and for every i, can we conclude that for every ? Note the assumption is only that the original basis is normal, nothing about orthogonality is assumed, otherwise this is trivial. I personally feel like it should just come from the transpose isomorphism, but he feels there might be something more subtle. If there's a reference or a really solid proof that actually has the maximization via the analysis definition taking the sup over unit vectors, that would probably be most convincing. Thanks, and any help appreciated.

No.

Take with , , , where is tiny. Then obviously (lifting the indices to show it is the dual), but has norm .

Excellent. Thanks a lot. He found that it was true for and it seems I was wrong to doubt that it must be true for all finite dimensional spaces.

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 Post subject: Re: Odd basic problemPosted: Wed, 25 Apr 2012 14:51:07 UTC
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outermeasure wrote:
A friend of mine posed the question if I have a basis of and a dual basis which has the usual pairing rule and for every i, can we conclude that for every ? Note the assumption is only that the original basis is normal, nothing about orthogonality is assumed, otherwise this is trivial. I personally feel like it should just come from the transpose isomorphism, but he feels there might be something more subtle. If there's a reference or a really solid proof that actually has the maximization via the analysis definition taking the sup over unit vectors, that would probably be most convincing. Thanks, and any help appreciated.

No.

Take with , , , where is tiny. Then obviously (lifting the indices to show it is the dual), but has norm .

Excellent. Thanks a lot. He found that it was true for and it seems I was wrong to doubt that it must be true for all finite dimensional spaces.

Well, for it is true because you have an (orientation-reversing) isometry swapping and , but it is not true in general that any permutation of is an isometry when in higher dimensions.

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