41. When studying vector spaces or related stuff in application, bilinear forms defined on them are often useful and indispensable. 42. The signature of this non-degenerate bilinear form being equal to the index of " X ". 43. In functional analysis a "'Dirichlet forms "'form a class of bilinear forms function spaces. 44. If is a real vector space, then we replace by its complexification and let denote the induced bilinear form on. 45. Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms. 46. Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms . 47. Since for an antiautomorphism we have for all in, if, then must be commutative and is a bilinear form . 48. Formally, the analogy is stated as a symmetric bilinear form ( multiplication ) and a quadratic form ( squaring ). 49. To do something like this in general, we can use any bilinear form , but that involves more structure than just 50. A "'quadratic Lie algebra "'is a Lie algebra together with a compatible symmetric bilinear form .
