Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

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 ... [Pg.577]

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... [Pg.577]

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... [Pg.89]

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... [Pg.12]

The inner product is a symmetric, bilinear operation if a, are real scalars and... [Pg.183]


See other pages where Scalar product bilinearity is mentioned: [Pg.12]    [Pg.28]    [Pg.39]    [Pg.41]    [Pg.37]    [Pg.236]    [Pg.496]    [Pg.421]    [Pg.300]   
See also in sourсe #XX -- [ Pg.126 ]




SEARCH



Bilinear

Scalar

© 2024 chempedia.info