On Completeness of Function Spaces

We first show that the set of all continuous functions C[a,b] is complete only under the uniform norm x oo = sup x t). Then we prove that the Fourier basis is 

The sequence space is a metric space since the metric axioms presented in Appendix C.4 are satisfied as follows  

Let u = tP - t = Referring to Fig.C.l, we understand that the rectangular area af is always smaller than the sum of the area (1) and the area (2). This results in 

By using (C.13), we obtain Holder s inequality (C.12). Note that the inequality for = = 2 in (C.12) is referred to as the Cauchy-Schwartz inequality.  

This result shows that is a metric space. 

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This form, which is analogous to that of Eq. (13), exhibits symbolically the fact that only the second part, i.e., a complex operator (matrix), represents the non-Hermitian nature of the problem in a complete function space of square-integrable functions, where the state-specific expectation value of H(r) is the real Eq and that of K re ) is the complex self-energy Translated into the language of function spaces, it is evident that diagonalization of the real H(r) on a space of real square-integrable functions yields ( Tq/Eo). It is then necessary to find a practical way to incorporate into the complete calculation the equivalent to the effects of the complex operator (matrix) K(rc ). 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

An intermolecular pair distribution function evaluated at the end of Step 2 would consist of delta functions at those distances allowed on the 2nnd lattice. After completion of reverse mapping, which moves the system from the discrete space of the lattice to a continuum, the carbon-carbon intermolecular pair distribution function becomes continuous, as depicted in Fig. 4.7 [144]. 

It is easy to see that we can form the matrix representation of any linear operator for any complete basis in any space. To do so, we act on each basis function with the operator and express the resulting function as a linear combination of the original basis functions. The coefficients that arise when we express the operator acting on the functions in terms of the original functions form the the matrix representation of the operator. 

Gdanitz and Ahlrichs devised a simpler variant of CPF, the averaged coupled-pair functional (ACPF) approach [30]. This produces results very similar to CPF for well-behaved closed-shell cases and is completely invariant to a unitary transformation on the occupied MOs. Its big advantage is that it can be cast in a multireference form. Multireference ACPF is probably the most sophisticated treatment of the correlation problem currently available that can be applied fairly widely, although it can encounter difficulties with the selection of reference spaces, as discussed elsewhere. 

It is much easier to introduce the complete basis in the space of functions depending on the spin variable of one electron. Allowable values of the spin projection s (in the units of K) are 1/2. Corresponding functions have the form ... 

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