International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 25-28   | 1 | 2 |

Section 1.3.2.2. Preliminary notions and notation

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.2. Preliminary notions and notation

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Throughout this text, [{\bb R}] will denote the set of real numbers, [{\bb Z}] the set of rational (signed) integers and [ {\bb N}] the set of natural (unsigned) integers. The symbol [{\bb R}^{n}] will denote the Cartesian product of n copies of [{\bb R}]:[{\bb R}^{n} = {\bb R} \times \ldots \times {\bb R} \quad (n \hbox{ times}, n \geq 1),]so that an element x of [{\bb R}^{n}] is an n-tuple of real numbers:[{\bf x} = (x_{1}, \ldots, x_{n}).]Similar meanings will be attached to [{\bb Z}^{n}] and [{\bb N}^{n}].

The symbol [{\bb C}] will denote the set of complex numbers. If [z \in {\bb C}], its modulus will be denoted by [|z|], its conjugate by [\bar{z}] (not [z^{*}]), and its real and imaginary parts by [{\scr Re}\, (z)] and [{\scr Im}\, (z)]:[{\scr Re}\, (z) = {\textstyle{1 \over 2}} (z + \bar{z}), \qquad {\scr Im}\, (z) = {1 \over 2i} (z - \bar{z}).]

If X is a finite set, then [|X|] will denote the number of its elements. If mapping f sends an element x of set X to the element [f(x)] of set Y, the notation[f: x \,\longmapsto\, f(x)]will be used; the plain arrow → will be reserved for denoting limits, as in[\lim\limits_{\rho \rightarrow \infty} \left(1 + {x \over p}\right)^{p} = e^{x}.]

If X is any set and S is a subset of X, the indicator function [\chi_{s}] of S is the real-valued function on X defined by[\eqalign{\chi_{S} (x) &= 1 \quad \hbox{if } x \in S\cr &= 0 \quad \hbox{if } x \,\notin\, S.}]

1.3.2.2.1. Metric and topological notions in [{\bb R}^{n}]

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The set [{\bb R}^{n}] can be endowed with the structure of a vector space of dimension n over [{\bb R}], and can be made into a Euclidean space by treating its standard basis as an orthonormal basis and defining the Euclidean norm:[\|{\bf x}\| = \left({\textstyle\sum\limits_{i = 1}^{n}} x_{i}^{2}\right)^{1/2}.]

By misuse of notation, x will sometimes also designate the column vector of coordinates of [{\bf x} \in {\bb R}^{n}]; if these coordinates are referred to an orthonormal basis of [{\bb R}^{n}], then[\|{\bf x}\| = ({\bf x}^{T} {\bf x})^{1/2},]where [{\bf x}^{T}] denotes the transpose of x.

The distance between two points x and y defined by [d({\bf x},{\bf y}) = \|{\bf x} - {\bf y}\|] allows the topological structure of [{\bb R}] to be transferred to [{\bb R}^{n}], making it a metric space. The basic notions in a metric space are those of neighbourhoods, of open and closed sets, of limit, of continuity, and of convergence (see Section 1.3.2.2.6.1[link]).

A subset S of [{\bb R}^{n}] is bounded if sup [\|{\bf x} - {\bf y}\| \,\lt\, \infty] as x and y run through S; it is closed if it contains the limits of all convergent sequences with elements in S. A subset K of [{\bb R}^{n}] which is both bounded and closed has the property of being compact, i.e. that whenever K has been covered by a family of open sets, a finite subfamily can be found which suffices to cover K. Compactness is a very useful topological property for the purpose of proof, since it allows one to reduce the task of examining infinitely many local situations to that of examining only finitely many of them.

1.3.2.2.2. Functions over [{\bb R}^{n}]

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Let ϕ be a complex-valued function over [{\bb R}^{n}]. The support of ϕ, denoted Supp ϕ, is the smallest closed subset of [{\bb R}^{n}] outside which ϕ vanishes identically. If Supp ϕ is compact, ϕ is said to have compact support.

If [{\bf t} \in {\bb R}^{n}], the translate of ϕ by t, denoted [\tau_{\bf t} \varphi], is defined by[(\tau_{\bf t} \varphi) ({\bf x}) = \varphi ({\bf x} - {\bf t}).]Its support is the geometric translate of that of ϕ:[\hbox{Supp } \tau_{\bf t} \varphi = \{{\bf x} + {\bf t} | {\bf x} \in \hbox{Supp } \varphi\}.]

If A is a nonsingular linear transformation in [{\bb R}^{n}], the image of ϕ by A, denoted [A^{\#} \varphi], is defined by[(A^{\#} \varphi) ({\bf x}) = \varphi [A^{-1} ({\bf x})].]Its support is the geometric image of Supp ϕ under A:[\hbox{Supp } A^{\#} \varphi = \{A ({\bf x}) | {\bf x} \in \hbox{Supp } \varphi\}.]

If S is a nonsingular affine transformation in [{\bb R}^{n}] of the form[S({\bf x}) = A({\bf x}) + {\bf b}]with A linear, the image of ϕ by S is [S^{\#} \varphi = \tau_{\bf b} (A^{\#} \varphi)], i.e.[(S^{\#} \varphi) ({\bf x}) = \varphi [A^{-1} ({\bf x} - {\bf b})].]Its support is the geometric image of Supp ϕ under S:[\hbox{Supp } S^{\#} \varphi = \{S({\bf x}) | {\bf x} \in \hbox{Supp } \varphi\}.]

It may be helpful to visualize the process of forming the image of a function by a geometric operation as consisting of applying that operation to the graph of that function, which is equivalent to applying the inverse transformation to the coordinates x. This use of the inverse later affords the `left-representation property' [see Section 1.3.4.2.2.2[link](e)[link]] when the geometric operations form a group, which is of fundamental importance in the treatment of crystallographic symmetry (Sections 1.3.4.2.2.4[link], 1.3.4.2.2.5[link]).

1.3.2.2.3. Multi-index notation

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When dealing with functions in n variables and their derivatives, considerable abbreviation of notation can be obtained through the use of multi-indices.

A multi-index [{\bf p} \in {\bb N}^{n}] is an n-tuple of natural integers: [{\bf p} = (p_{1}, \ldots, p_{n})]. The length of p is defined as[|{\bf p}| = {\textstyle\sum\limits_{i = 1}^{n}}\, p_{i},]and the following abbreviations will be used:[\displaylines{\quad (\hbox{i})\qquad\,\,{\bf x}^{{\bf p}} = x_{1}^{p_{1}} \ldots x_{n}^{p_{n}}\hfill\cr \quad (\hbox{ii})\,\qquad D_{i\,} f = {\partial f \over \partial x_{i}} = \partial_{i}\, f\hfill\cr \quad (\hbox{iii})\qquad D^{{\bf p}} f = D_{1}^{p_{1}} \ldots D_{n}^{p_{n}} f = {\partial^{|{\bf p}|} f \over \partial x_{1}^{p_{1}} \ldots \partial x_{n}^{p_{n}}}\hfill\cr \quad (\hbox{iv})\qquad {\bf q} \leq {\bf p} \hbox{ if and only if } q_{i} \leq p_{i} \hbox{ for all } i = 1, \ldots, n\hfill\cr \quad (\hbox{v})\qquad\,{\bf p} - {\bf q} = (p_{1} - q_{1}, \ldots, p_{n} - q_{n})\hfill\cr \quad (\hbox{vi})\qquad {\bf p}! = p_{1}! \times \ldots \times p_{n}!\hfill\cr \quad (\hbox{vii})\qquad\!\! \pmatrix{{\bf p}\cr {\bf q}\cr} = \pmatrix{p_{1}\cr q_{1}\cr} \times \ldots \times \pmatrix{p_{n}\cr q_{n}\cr}.\hfill}]

Leibniz's formula for the repeated differentiation of products then assumes the concise form[D^{\bf p} (fg) = \sum\limits_{{\bf q} \leq {\bf p}} \pmatrix{{\bf p}\cr {\bf q}\cr} D^{{\bf p} - {\bf q}} f D^{\bf q} g,]while the Taylor expansion of f to order m about [{\bf x} = {\bf a}] reads[f({\bf x}) = \sum\limits_{|{\bf p}| \leq m} {1 \over {\bf p}!} [D^{\bf p} f ({\bf a})] ({\bf x} - {\bf a})^{\bf p} + o (\|{\bf x} - {\bf a}\|^{m}).]

In certain sections the notation [\nabla f] will be used for the gradient vector of f, and the notation [(\nabla \nabla^{T})f] for the Hessian matrix of its mixed second-order partial derivatives:[\displaylines{\nabla = \pmatrix{\displaystyle{\partial \over \partial x_{1}}\cr \vdots\cr\noalign{\vskip6pt} {\displaystyle{\partial \over \partial x_{n}}}\cr}, \quad \nabla f = \pmatrix{\displaystyle{\partial f \over \partial x_{1}}\cr \vdots\cr\noalign{\vskip6pt} {\displaystyle{\partial f \over \partial x_{n}}}\cr},\cr (\nabla \nabla^{T}) f = \pmatrix{\displaystyle{\partial^{2} f \over \partial x_{1}^{2}} &\ldots &{\displaystyle{\partial^{2} f \over \partial x_{1} \partial x_{n}}}\cr \vdots &\ddots &\vdots\cr\noalign{\vskip6pt} {\displaystyle{\partial^{2} f \over \partial x_{n} \partial x_{1}}} &\ldots &{\displaystyle{\partial^{2} f \over \partial x_{n}^{2}}}\cr}.}]

1.3.2.2.4. Integration, [L^{p}] spaces

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The Riemann integral used in elementary calculus suffers from the drawback that vector spaces of Riemann-integrable functions over [{\bb R}^{n}] are not complete for the topology of convergence in the mean: a Cauchy sequence of integrable functions may converge to a nonintegrable function.

To obtain the property of completeness, which is fundamental in functional analysis, it was necessary to extend the notion of integral. This was accomplished by Lebesgue [see Berberian (1962)[link], Dieudonné (1970)[link], or Chapter 1 of Dym & McKean (1972)[link] and the references therein, or Chapter 9 of Sprecher (1970)[link]], and entailed identifying functions which differed only on a subset of zero measure in [{\bb R}^{n}] (such functions are said to be equal `almost everywhere'). The vector spaces [L^{p} ({\bb R}^{n})] consisting of function classes f modulo this identification for which[\|{\bf f}\|_{p} = \left({\textstyle\int\limits_{{\bb R}^{n}}} |\,f ({\bf x}) |^{p}\ {\rm d}^{n} {\bf x}\right)^{1/p} \,\lt\, \infty]are then complete for the topology induced by the norm [\|.\|_{p}]: the limit of every Cauchy sequence of functions in [L^{p}] is itself a function in [L^{p}] (Riesz–Fischer theorem).

The space [L^{1} ({\bb R}^{n})] consists of those function classes f such that[\|\,f \|_{1} = {\textstyle\int\limits_{{\bb R}^{n}}} |\,f ({\bf x})|\,\hbox{d}^{n} {\bf x} \,\lt\, \infty]which are called summable or absolutely integrable. The convolution product:[\eqalign{(\,f * g) ({\bf x}) &= {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf y}) g({\bf x} - {\bf y})\,\hbox{d}^{n} {\bf y}\cr &= {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x} - {\bf y}) g ({\bf y})\,\hbox{d}^{n} {\bf y} = (g * f) ({\bf x})}]is well defined; combined with the vector space structure of [L^{1}], it makes [L^{1}] into a (commutative) convolution algebra. However, this algebra has no unit element: there is no [f \in L^{1}] such that [f * g = g] for all [g \in L^{1}]; it has only approximate units, i.e. sequences [(f_{\nu })] such that [f_{\nu } * g] tends to g in the [L^{1}] topology as [\nu \rightarrow \infty]. This is one of the starting points of distribution theory.

The space [L^{2} ({\bb R}^{n})] of square-integrable functions can be endowed with a scalar product[(\,f, g) = {\textstyle\int\limits_{{\bb R}^{n}}} \overline{f({\bf x})} g({\bf x})\,\hbox{d}^{n} {\bf x}]which makes it into a Hilbert space. The Cauchy–Schwarz inequality[|(\,f, g)| \leq [(\,f, f) (g, g)]^{1/2}]generalizes the fact that the absolute value of the cosine of an angle is less than or equal to 1.

The space [L^{\infty} ({\bb R}^{n})] is defined as the space of functions f such that[\|\,f \|_{\infty} = \lim\limits_{p \rightarrow \infty} \|\,f \|_{p} = \lim\limits_{p \rightarrow \infty} \left({\textstyle\int\limits_{{\bb R}^{n}}} |\, f({\bf x}) |^{p} \,\hbox{d}^{n} {\bf x}\right)^{1/p} \,\lt\, \infty.]The quantity [\|\,f \|_{\infty}] is called the `essential sup norm' of f, as it is the smallest positive number which [|\,f({\bf x})|] exceeds only on a subset of zero measure in [{\bb R}^{n}]. A function [f \in L^{\infty}] is called essentially bounded.

1.3.2.2.5. Tensor products. Fubini's theorem

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Let [f \in L^{1} ({\bb R}^{m})], [g \in L^{1} ({\bb R}^{n})]. Then the function[f \otimes g: ({\bf x},{\bf y}) \,\longmapsto\, f({\bf x}) g({\bf y})]is called the tensor product of f and g, and belongs to [L^{1} ({\bb R}^{m} \times {\bb R}^{n})]. The finite linear combinations of functions of the form [f \otimes g] span a subspace of [L^{1} ({\bb R}^{m} \times {\bb R}^{n})] called the tensor product of [L^{1} ({\bb R}^{m})] and [L^{1} ({\bb R}^{n})] and denoted [L^{1} ({\bb R}^{m}) \otimes L^{1} ({\bb R}^{n})].

The integration of a general function over [{\bb R}^{m} \times {\bb R}^{n}] may be accomplished in two steps according to Fubini's theorem. Given [F \in L^{1} ({\bb R}^{m} \times {\bb R}^{n})], the functions[\eqalign{F_{1} : {\bf x} &\,\longmapsto\, {\textstyle\int\limits_{{\bb R}^{n}}} F ({\bf x},{\bf y}) \,\hbox{d}^{n} {\bf y}\cr F_{2} : {\bf y} &\,\longmapsto\, {\textstyle\int\limits_{{\bb R}^{m}}} F ({\bf x},{\bf y}) \,\hbox{d}^{m} {\bf x}}]exist for almost all [{\bf x} \in {\bb R}^{m}] and almost all [{\bf y} \in {\bb R}^{n}], respectively, are integrable, and[\textstyle\int\limits_{{\bb R}^{m} \times {\bb R}^{n}} F ({\bf x},{\bf y}) \,\hbox{d}^{m} {\bf x} \,\hbox{d}^{n} {\bf y} = {\textstyle\int\limits_{{\bb R}^{m}}} F_{1} ({\bf x}) \,\hbox{d}^{m} {\bf x} = {\textstyle\int\limits_{{\bb R}^{n}}} F_{2} ({\bf y}) \,\hbox{d}^{n} {\bf y}.]Conversely, if any one of the integrals[\displaylines{\quad (\hbox{i})\qquad {\textstyle\int\limits_{{\bb R}^{m} \times {\bb R}^{n}}} |F ({\bf x},{ \bf y})| \,\hbox{d}^{m} {\bf x} \,\hbox{d}^{n} {\bf y}\qquad \hfill\cr \quad (\hbox{ii})\qquad {\textstyle\int\limits_{{\bb R}^{m}}} \left({\textstyle\int\limits_{{\bb R}^{n}}} |F ({\bf x},{ \bf y})| \,\hbox{d}^{n} {\bf y}\right) \,\hbox{d}^{m} {\bf x}\hfill\cr \quad (\hbox{iii})\qquad {\textstyle\int\limits_{{\bb R}^{n}}} \left({\textstyle\int\limits_{{\bb R}^{m}}} |F ({\bf x},{ \bf y})| \,\hbox{d}^{m} {\bf x}\right) \,\hbox{d}^{n} {\bf y}\hfill}]is finite, then so are the other two, and the identity above holds. It is then (and only then) permissible to change the order of integrations.

Fubini's theorem is of fundamental importance in the study of tensor products and convolutions of distributions.

1.3.2.2.6. Topology in function spaces

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Geometric intuition, which often makes `obvious' the topological properties of the real line and of ordinary space, cannot be relied upon in the study of function spaces: the latter are infinite-dimensional, and several inequivalent notions of convergence may exist. A careful analysis of topological concepts and of their interrelationship is thus a necessary prerequisite to the study of these spaces. The reader may consult Dieudonné (1969[link], 1970[link]), Friedman (1970)[link], Trèves (1967)[link] and Yosida (1965)[link] for detailed expositions.

1.3.2.2.6.1. General topology

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Most topological notions are first encountered in the setting of metric spaces. A metric space E is a set equipped with a distance function d from [E \times E] to the non-negative reals which satisfies:[\matrix{(\hbox{i})\hfill & d(x, y) = d(y, x)\hfill &\forall x, y \in E\hfill &\hbox{(symmetry)\semi}\hfill\cr\cr (\hbox{ii})\hfill &d(x, y) = 0 \hfill &\hbox{iff } x = y\hfill &\hbox{(separation)\semi}\hfill\cr\cr (\hbox{iii})\hfill & d(x, z) \leq d(x, y) + d(y, z)\hfill &\forall x, y, z \in E\hfill &\hbox{(triangular}\hfill\cr& & &\hbox{inequality).}\hfill}]By means of d, the following notions can be defined: open balls, neighbourhoods; open and closed sets, interior and closure; convergence of sequences, continuity of mappings; Cauchy sequences and completeness; compactness; connectedness. They suffice for the investigation of a great number of questions in analysis and geometry (see e.g. Dieudonné, 1969[link]).

Many of these notions turn out to depend only on the properties of the collection [{\scr O}(E)] of open subsets of E: two distance functions leading to the same [{\scr O}(E)] lead to identical topological properties. An axiomatic reformulation of topological notions is thus possible: a topology in E is a collection [{\scr O}(E)] of subsets of E which satisfy suitable axioms and are deemed open irrespective of the way they are obtained. From the practical standpoint, however, a topology which can be obtained from a distance function (called a metrizable topology) has the very useful property that the notions of closure, limit and continuity may be defined by means of sequences. For nonmetrizable topologies, these notions are much more difficult to handle, requiring the use of `filters' instead of sequences.

In some spaces E, a topology may be most naturally defined by a family of pseudo-distances [(d_{\alpha})_{\alpha \in A}], where each [d_{\alpha}] satisfies (i) and (iii) but not (ii). Such spaces are called uniformizable. If for every pair [(x, y) \in E \times E] there exists [\alpha \in A] such that [d_{\alpha} (x, y) \neq 0], then the separation property can be recovered. If furthermore a countable subfamily of the [d_{\alpha}] suffices to define the topology of E, the latter can be shown to be metrizable, so that limiting processes in E may be studied by means of sequences.

1.3.2.2.6.2. Topological vector spaces

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The function spaces E of interest in Fourier analysis have an underlying vector space structure over the field [{\bb C}] of complex numbers. A topology on E is said to be compatible with a vector space structure on E if vector addition [i.e. the map [({\bf x},{ \bf y}) \,\longmapsto\, {\bf x} + {\bf y}]] and scalar multiplication [i.e. the map [(\lambda, {\bf x}) \,\longmapsto\, \lambda {\bf x}]] are both continuous; E is then called a topological vector space. Such a topology may be defined by specifying a `fundamental system S of neighbourhoods of [{\bf 0}]', which can then be translated by vector addition to construct neighbourhoods of other points [{\bf x} \neq {\bf 0}].

A norm ν on a vector space E is a non-negative real-valued function on [E \times E] such that[\displaylines{\quad (\hbox{i}')\,\,\quad\nu (\lambda {\bf x}) = |\lambda | \nu ({\bf x}) \phantom{|\lambda | v ({\bf x} =i} \hbox{for all } \lambda \in {\bb C} \hbox{ and } {\bf x} \in E\hbox{\semi}\hfill\cr \quad (\hbox{ii}')\,\quad\nu ({\bf x}) = 0 \phantom{|\lambda | v ({\bf x} = |\lambda | vxxx} \hbox{if and only if } {\bf x} = {\bf 0}\hbox{\semi}\hfill\cr \quad (\hbox{iii}')\quad \nu ({\bf x} + {\bf y}) \leq \nu ({\bf x}) + \nu ({\bf y}) \quad \hbox{for all } {\bf x},{\bf y} \in E.\hfill}]Subsets of E defined by conditions of the form [\nu ({\bf x}) \leq r] with [r\,\gt\, 0] form a fundamental system of neighbourhoods of 0. The corresponding topology makes E a normed space. This topology is metrizable, since it is equivalent to that derived from the translation-invariant distance [d({\bf x},{ \bf y}) = \nu ({\bf x} - {\bf y})]. Normed spaces which are complete, i.e. in which all Cauchy sequences converge, are called Banach spaces; they constitute the natural setting for the study of differential calculus.

A semi-norm σ on a vector space E is a positive real-valued function on [E \times E] which satisfies (i′) and (iii′) but not (ii′). Given a set Σ of semi-norms on E such that any pair (x, y) in [E \times E] is separated by at least one [\sigma \in \Sigma], let B be the set of those subsets [\Gamma_{\sigma{, \,} r}] of E defined by a condition of the form [\sigma ({\bf x}) \leq r] with [\sigma \in \Sigma] and [r \,\gt\, 0]; and let S be the set of finite intersections of elements of B. Then there exists a unique topology on E for which S is a fundamental system of neighbourhoods of 0. This topology is uniformizable since it is equivalent to that derived from the family of translation-invariant pseudo-distances [({\bf x},{ \bf y}) \,\longmapsto\] [ \sigma ({\bf x} - {\bf y})]. It is metrizable if and only if it can be constructed by the above procedure with Σ a countable set of semi-norms. If furthermore E is complete, E is called a Fréchet space.

If E is a topological vector space over [{\bb C}], its dual [E^{*}] is the set of all linear mappings from E to [{\bb C}] (which are also called linear forms, or linear functionals, over E). The subspace of [E^{*}] consisting of all linear forms which are continuous for the topology of E is called the topological dual of E and is denoted E′. If the topology on E is metrizable, then the continuity of a linear form [T \in E'] at [f \in E] can be ascertained by means of sequences, i.e. by checking that the sequence [[T(\,f_{j})]] of complex numbers converges to [T(\,f)] in [{\bb C}] whenever the sequence [(\,f_{j})] converges to f in E.

References

Berberian, S. K. (1962). Measure and Integration. New York: Macmillan. [Reprinted by Chelsea, New York, 1965.]
Dieudonné, J. (1969). Foundations of Modern Analysis. New York, London: Academic Press.
Dieudonné, J. (1970). Treatise on Analysis, Vol. II. New York, London: Academic Press.
Dym, H. & McKean, H. P. (1972). Fourier Series and Integrals. New York, London: Academic Press.
Friedman, A. (1970). Foundations of Modern Analysis. New York: Holt, Rinehart & Winston. [Reprinted by Dover, New York, 1982.]
Sprecher, D. A. (1970). Elements of Real Analysis. New York: Academic Press. [Reprinted by Dover Publications, New York, 1987.]
Trèves, F. (1967). Topological Vector Spaces, Distributions, and Kernels. New York, London: Academic Press.
Yosida, K. (1965). Functional Analysis. Berlin: Springer-Verlag.








































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