Inner product space

The round brackets indicate the so-called /i-product which is a generalisation of the usual Hilbert space inner product with an indefinite metric. The extended operator H takes the role of an excitation energy operator which lives, like the extended states A,B), in an extended Hilbert space Y. These objects will be defined rigorously in this chapter in a very formal manner. In first reading these formal definitions may well be skipped. 

According to the Helmholtz theorem the Hilbert space of 2-D vector fields p x, y) with the inner product... 

This set forms a Hilbert space with an inner product defined by... 

Linear bounded operators in a real Hilbert space. Let H he a real Hilbert space equipped with an inner product x,y) and associated norm II X II = (x, x). We consider bounded linear operators defined on the space... 

Let A be a positive self-adjoint linear operator. By introducing on the space H the inner product x,y) = Ax,y) and the associated norm x) we obtain a Hilbert space Ha, which is usually called the energetic space Ha- It is easy to show that the inner product... 

Linear operators in finite-dimensional spaces. It is supposed that an n-dimensional vector space R is equipped with an inner product (, ) and associated norm a = / x x). By the definition of finite-dimensional space, any vector x G i n can uniquely be represented as a linear combina-... 

In what follows we will use, as a rule, several norms associated with inner products in the space Hh (the grid analogs of the L2 and IT2-norms are available in Chapter 1). 

We will assume that problem (37) is solvable for any right-hand sides (p H there exists an operator A with the domain V A ) = H. All the constants below are supposed to be independent of h. In what follows the space H is equipped with an inner product (, ) and associated norm II. T II = /i x, x ). The writing A = A > 0 means that A is a self-adjoint... 

Let T be a self-adjoint positive definite linear operator in Hilbert space H equipped with an inner product (,) and let / be a given element of the space H. The problem of minimizing the functional... 

In the case of the second eigenvalue boundary-value problem (8) the space Hh = Clh comprises all the functions defined on the grid ujh, the inner product (,) on Hh is to be understood in the sense (14) and the operator A is defined as a sum... 

One obvious way of proceeding is to introduce the space H of all grid functions defined on and vanishing for i = N. Under the inner product structure... 

The inner product in the space Q /j of all grid vector-functions given on the grid ijjp and vanishing on the boundary is defined by... 

Ay — f, where Ay = —Ay in the space H = Cl of all grid functions given on the grid W/j and vanishing on its boundary An inner product and associated norm in that space H are defined by... 

Appendix Proof of the inner product hermiticity of a subspace of an hermitian space... 

Our first way of answering the last question will be based on the fundamental theorems on Hilbert space [14], Indeed, the theorem on separability tells us that any subspace of h is also a separable Hilbert space. As a consequence, the inner product defined on, say, the occupied subspace is hermitian irrespectively of the choice of the basis x f (/)], as long as this latter satisfies the fundamental requirements of Quantum Mechanics. One should therefore not have to impose this property as a constraint when counting the number of conditions arising from the constraint CC+ =1 but, on the contrary, can take it for granted. 

A vector space L defined over a field F is further called an inner-product space or unitary space if its elements satisfy one more condition ... 

The linear space of all n-tuplets of complex numbers becomes an inner-product space if the scalar product of the two elements u and v is defined as the complex number given by... 

The ordinary three-dimensional space of position vectors is also an inner product space with the familiar rule for taking the scalar product of two vectors. 

Consider a group whose elements can be put in one-to-one correspondence with the points of a subset of an n-dimensional real inner-product space Sn, as parameter space. Let x and x2 be any two elements of a group G with images P(x ) and P(x 2) in Sn. If it is possible to connect P x ) and P x2) by one or more paths lying entirely within the parameter space, the parameter space is said to be connected otherwise it is disconnected. 

The overlap integrals form the inner products of the linear space of the AOs 0j. Due to a confusion between the two roles of differentials, the matrix S is sometimes called the metric of the linear space. A metric m involving the 0i must satisfy m(0 , 03) < m(0i, 02) + m(02,03) and m(0i, 02) = 0 (i = 02), hence m(0i, 02) > 0. Clearly the overlap matrix satisfies none of these requirements. A genuine metric can be defined in terms of S the matrix M = Z - S satisfies the above axioms where Z is a matrix containing unity in every position. 

Note the similarity of Eqs. 2.43 and 2.44 with Eqs. 2.80 and 2.81 because both the vectors in the former equations and the functions of the latter are all elements of linear vector spaces. The main difference arises in the way in which the inner products are evaluated. Also, as was the case for vectors, if the field functions are non-negative functions, SCar(F, F pj will be non-negative. When this is not the case, however, Sr.ir(F (,F g) may become negative, a situation that also obtains for the other similarity indices discussed in the remainder of this section. Maggiora et al. (43) have treated this case in great detail for continuous field functions, but the arguments can be carried through for finite vectors as well (vide supra). 

The inner product of two orbitals is sometimes called the overlap, the name suggesting the extent to which the orbitals occupy the same region of space it provides a measure of how closely two functions resemble each other. In particular ... 

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. 

---