Scalar product bilinearity

The indefinite bilinear form (XF, TB) as defined by Eq. (9-687) can be expressed in terms of the Hilbert space scalar product (Y,x)h 85 follows ... 

In effect the scalar product in (9-688), which makes the vector space into a Hilbert space, omits the factor ( —1) from the bilinear form (9-687). We shall always work with the indefinite bilinear form (9-687). Thus, for example, one verifies that with this indefinite metric... 

Does this T have the desired property By the bilinearity of the complex scalar products (condition 1 of Definition 3.2), it suffices to check the condition on basis elements. For any j and any k we have... 

A highly important role in geometry in played by Riemannian manifolds, that is, smooth n-dimensional manifolds M supplied with a Riemannian metric It is known that setting this metric is equivalent to setting in each tangent space TxM at a point x G M a, bilinear symmetric nondegenerate positive definite scalar product which smoothly depends on the point x. V a, b G TxM is an arbitrary pair of tangent vectors, this scalar product can be written in the form... 

The inner product is a symmetric, bilinear operation if a, are real scalars and... 

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