Scalar product definition

Krambeck (1994b) observes that the scalar product definition should include the number den-... 

Likewise, the scalar products between submatrices of the TDQSH, as defined in Eq. (22), entering the metric matrices similar to the Eq. (23), can be easily employed to construct a new (MxM) matrix constituted by bimolecular CSI-like elements. This new CSI matrix construction can be achieved by using the already presented scalar product definition, involving the submatrices as defined earlier in Eq. (22), that is ... 

Other postulates required to complete the definition of will not be listed here they are concerned with the existence of a basis set of vectors and we shall discuss that question in some detail in the next section. For the present we may summarize the above defining properties of Hilbert space by saying that it is a linear space with a complex-valued scalar product. 

Angular distance or angle between two points x and y, as seen from the origin of space, is derived from the definition of the scalar product in terms of the norms of the vectors ... 

Note that in data analysis we divide by n in the definition of standard deviation rather than by the factor n - 1 which is customary in statistical inference. Likewise we can relate the product-moment (or Pearson) coefficient of correlation r (Section 8.3.1) to the scalar product of the vectors (x - x) and (y - y) ... 

The significance of this notation is discussed in Section 3.6. From the definition of the scalar product and of the notation (0 0), we note that... 

Transformations that take one orthonormal set of basis vectors into another orthonormal set are called unitary transformations, the operators associated with them are called unitary operators. This definition preserves the norms and scalar products of vectors in Ln. The transformation (4) is in fact a set of linear equations... 

Definition 15 A -body operator is a Hermitian operator that can be represented as a polynomial of degree 2 A in the annihilation and creation operators, and is of even degree in these operators. In addition, a A -body operator must be orthogonal to all k — l)-body operators, all k — 2)-body operators,. .., and all scalar operators, with respect to the trace scalar product. 

Because all of the components of J are Hermitian, and because the scalar product of any function with itself is positive semi-definite, the following identity holds ... 

Physicists often refer to complex scalar product spaces as Hilbert spaces. The formal mathematical definition of a Hilbert space requires more than just the existence of a complex scalar product the space must be closed a.k.a. complete in a certain technical sense. Because every scalar product space is a subset of some Hilbert space, the discrepancy in terminology between mathematicians and physicists does not have dire consequences. However, in this text, to avoid discrepancies with other mathematics textbooks, we will use complex scalar product. ... 

We start with the definition of a complex scalar product (also known as a Hermitian inner product, a complex inner product or a unitary structure on a complex vector space. Then we present several examples of complex scalar product spaces. 

Since a complex scalar product resembles the EucUdean dot product in its form and definition, we can use our intuition about perpendicularity in the Euclidean three-space we inhabit to study complex scalar product spaces. However, we must be aware of two important differences. Eirst, we are dealing with complex scalars rather than real scalars. Second, we are often dealing with infinite-dimensional spaces. It is easy to underestimate the trouble that infinite dimensions can cause. If this section seems unduly technical (especially the introduction to orthogonal projections), it is because we are careful to avoid the infinite-dimensional traps. 

Definition 3.6 Suppose B is an arbitrary subset of a complex scalar product space V. Then the perpendicular space to B in V is... 

If V is finite dimensional, then Definition 3.7 is consistent with Definition 2.2 (Exercise 3.13). In infinite-dimensional complex scalar product spaces. Definition 3.7 is usually simpler than an infinite-dimensional version of Definition 2.2. To make sense of an infinite linear combination of functions, one must address issues of convergence however, arguments involving perpendicular subspaces are often relatively simple. We can now define unitary bases. 

Definition 3.8 Suppose V is a complex scalar product space and B is a subset ofV. Suppose that B satisfies the following ... 

Definition 3.9 Suppose V and W are finite-dimensional complex scalar product spaces, and let ( , > v and , denote their complex scalar products. Suppose T.V W is a linear transformation, that is, suppose T Hom (V, VP). Then the adjoint of T is the unique linear transformation T W V such that for all v e V and all w e W we have... 

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... 

Definition 3.11 Suppose V is a complex scalar product space. An orthogonal projection IT V V is a Hemitian linear operator FI such that IT = fl. 

The goal of this section is to find useful spanning subspaces of C[— 1, 1] and 2(52) Recall from Definition 3.7 that a subspace spans if the perpendicular subspace is trivial. In a finite-dimensional space V, there are no proper spanning subspaces any subspace that spans must have the same dimension as V and hence is equal to V. However, for an infinite-dimensional complex scalar product space the situation is more complicated. There are often proper subspaces that span. We will see that polynomials span both C[—l, 1] andL2(5 2) in Propositions 3.8 and 3.9, respectively. In the process, we will appeal to the Stone-Weierstrass theorem (Theorem 3.2) without giving its proof. 

Exercise 3.13 In this exercise V is a finite-dimensional complex scalar product space and W is a subspace ofV. Show that VP- - = Q in V if and only if VK spans V in the sense given in Definition 2.2. 

For each nonnegative integer f, the space of spherical harmonics of degree f (see Dehnition 2.6) is the vector space for a representation of 50(3). These representations appear explicitly in our analysis of the hydrogen atom in Chapter 7. Recall the complex scalar product space L (S ) from Definition 3.3. 

With this complex scalar product on the dual space V in hand, we can make the relationship between the dual and the adjoint clear. The definition... 

We start by defining the projective unitary representations. Recall the unitary group ZT (V) of a complex scalar product space V from Definition 4.2. The following definition is an analog of Definition 4.11. 

Definition 10.7 Suppose G is a group and V is a complex scalar product space. Then the triple G, V, p) is called a projective unitary representation if and only if p is a group homomorphism from G to PIT (V). 

Definition 10.8 Suppose G is a group, V is a complex scalar product space and p. G PU (V) is a projective unitary representation. We say that p is irreducible if the only subspace W of V such that [VT] is invariant under p is V itself. 

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