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Vectors

Consider a vector space V over some field F.

My question is that is the inner product in the specified vector field unique?

I mean can there be more than one mapping that satisfy all axioms required for that mapping to be an inner product?

What about R^3 Euclidean vector space.Is the dot product the only inner product possible in euclidean vector space?
(edited 9 years ago)
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Andy98
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Original post by Many-Faced God
Consider a vector space V over some field F.

My question is that is the inner product in the specified vector field unique?

I mean can there be more than one mapping that satisfy all axioms required for that mapping to be an inner product?

What about R^3 Euclidean vector space.Is the dot product the only inner product possible in euclidean vector space?


Honestly, I'm not sure on that one. I'll ask someone who might know.
The inner product is not necessarily unique. And I believe intuitively that the only space with a unique inner product is the trivial space (sometimes called the point space).


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(edited 9 years ago)
Original post by LightBlueSoldier
The inner product is not necessarily unique. And I believe intuitively that the only space with a unique inner product is the trivial space (sometimes called the point space).


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Thank you for your reply
What about in the euclidean space.Is it unique in the euclidean space.If not can you provide an example of such an inner product in the R^3 euclidean space that is not the dot product.


I think for odd dimensional euclidean spaces

the inner product defined by x∗yx*y = (−1i+1)(-1^{i+1})∑i=1nxiyi\displaystyle\sum_{i=1}^n x_iy_i satisfies all the conditions.

Could you verify that
(edited 9 years ago)
Original post by Many-Faced God
Thank you for your reply
What about in the euclidean space.Is it unique in the euclidean space.If not can you provide an example of such an inner product in the R^3 euclidean space that is not the dot product.


I think for odd dimensional euclidean spaces

the inner product defined by x∗yx*y = (−1i+1)(-1^{i+1})∑i=1nxiyi\displaystyle\sum_{i=1}^n x_iy_i satisfies all the conditions.

Could you verify that


A simple example like <x,y>=2x1y1+x2y2+x3y3<x,y>= 2 x_1 y_1 +x_2 y_2 +x_3 y_3 suffices.


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