What is the significance of equipping the vector space with a form? For instance a symplectic space has a symlectic two form? Why does it need it + what does having it allow/benifit us?
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To relate a vector space to its dual vector space I guess? – Troy Woo Dec 17 '14 at 10:15
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Bilinear forms on a vector space equip the vector space with a geometry. If all you have is a vector space, then you can't speak of angles or lengths of vectors. If you have a bilinear form (such as an inner product, or a symplectic form) then you can speak of various geometric concepts. So, roughly, vector space + a form = richer geometry.
Ittay Weiss
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