# Complex Hilbert Space as a Symplectic Space?

#### Bacle

Hi All: in the page:

http://mathworld.wolfram.com/SymplecticForm.html,

Complex Hilbert space, with "the inner-product" I<x,y> , where <.,.> is the inner-product

Does this refer to taking the imaginary part of the standard inner-product ? If so, is

I<x,y> symplectic in Complex Hilbert Space? It is obviously bilinear, but I don't see

how it is antisymmetric , i.e., I don't see that I<x,y>=-I<y,x>

Am I missing something?

Thanks.

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#### Fredrik

Staff Emeritus
Gold Member
For any complex number c=a+ib (with a,b real), we have Im(c*) = Im(a-ib) = -b = -Im(a+ib) = -Im c, so

Im<x,y>=Im(<y,x>*)=-Im<y,x>

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#### Bacle

Yes, how dumb of me. Thanks, Fredrik.