Index set (math)

In mathematics , an index (plural: indices ) denotes an element of an index set that is used to number a wide variety of objects. The set of natural numbers is often used as the index set .

In the early days of the mathematics of the modern age is also a function value was - f (x) in modern notation - means subscripted x as f _x referred to. The notation a _i for the terms of a sequence (as a function over natural numbers) can be seen as a holdover from this older notation. Depending on requirements, superscript indices a ^{i can also} occur despite the risk of confusion with the power calculation.

index

An index is a distinguishing mark that is attached to the top or bottom, right or left of a character.

In mathematics, the character to which the index is attached stands for a mathematical object and the index itself is preferably noted at the bottom right of this character. However, any other position in the index is also conceivable, depending on the mathematical subject and question.

Examples

• In the case of function families, the family parameters are usually noted as an index, while the "normal" arguments are written in brackets after the function name - e.g. B. ${\displaystyle f_{t}(x).}$

• In a matrix , its components, i.e. the individual values ​​in the matrix, are often indexed. For example, the component representation of a matrix is ${\displaystyle 3\times 3}$ ${\displaystyle A}$

${\displaystyle A={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}\,.}$

Each component has exactly two indices. The first index indicates in which row and the second in which column of the matrix the component is located. When not both indices consist of one symbol, many authors used a comma between them: . ${\displaystyle a_{ij}}$ ${\displaystyle a_{m+1,n+1}}$

• In physics, especially in the tensor representations of physics , double indices are used for the abbreviated notation of sums. This convention is called Einstein's sum convention .

• With stochastic processes and time series , the time parameter is often written as an index.

• In the mathematical branch of differential geometry , the sections of a vector bundle are often denoted in index notation in order to have free the functional notation for algebraic operations between fibers of different bundles over the same point.

• In the function theory of several variables denotes the ring of power series converging around 0 in indeterminate. The reason for writing the lower-left index is that the lower-right position is reserved for subsets in whose environment functions should be holomorphic. You then write so that you have a left and a right index. So is . ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle n}$ ${\displaystyle K\subset \mathbb {C} ^{n}}$ ${\displaystyle {}_{n}{\mathcal {O}}_{K}}$ ${\displaystyle {}_{n}{\mathcal {O}}={}_{n}{\mathcal {O}}_{\{0\}}}$

• In homological algebra , the indices below are used for constructions from chain complexes ; the indices above are used for the corresponding constructions from coquette complexes . Therefore, homology groups are designated with and cohomology groups with . ${\displaystyle H_{n}}$ ${\displaystyle H^{n}}$

Index amount

definition

A set whose elements index elements of another set is called an index set.

annotation

An index set is therefore not a special set, but rather that the elements of the set are used to index other objects. In many cases the set of natural numbers is used for this. However, any set, whether with a finite, countable or uncountable number of elements, can be used as an index set and then combines mathematical objects into a family (here is the index set). If the natural numbers are used as the index set, one speaks of a sequence instead of a family . The term sequence is also used for families indexed with ordinals . ${\displaystyle A_{\bullet }}$ ${\displaystyle (A_{\lambda })_{\lambda \in \Lambda }}$ ${\displaystyle \Lambda }$

Selection function

In mathematics, the index can be formally defined using the selection function as a mapping from the index set to the set of indexed objects.

definition

If there are any sets , then one can use the n-tuple ${\displaystyle X_{1},\ldots ,X_{n}}$

${\displaystyle x=(x_{1},\ldots ,x_{n})}$ With ${\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}}$

as picture

${\displaystyle x\colon \,\{1,\ldots ,n\}\rightarrow X_{1}\cup \ldots \cup X_{n},\,i\mapsto x(i)=:x_{i}\in X_{i}}$,

grasp. It is called a selection function. ${\displaystyle x}$

Axiom of choice

If one does not want to limit oneself to a finite number of sets , but to consider an infinite number (especially uncountable numbers), then the existence of the selection function just defined is not clear. This means that with infinitely large index sets it is not always possible to find a concrete representation for the selection function and thus to show the existence of this. The axiom of choice ensures that such a selection function does exist . However, the axiom says nothing about the concrete representation of the selection function. ${\displaystyle X_{i}}$

See also

• Multi-index

Web links

• Eric W. Weisstein : Index . In: MathWorld (English).
• Eric W. Weisstein : Index Set . In: MathWorld (English).

Individual evidence

1. ↑ index . In: FA Brockhaus (ed.): The new Brockhaus . All book in five volumes and an atlas. 3rd completely revised edition. tape 2 . Wiesbaden 1962, p. 608 , col. 2 .
2. ↑ Gerd Fischer : Linear Algebra. (An introduction for first-year students). 14th revised edition. Vieweg-Verlag, Wiesbaden 2003, ISBN 3-528-03217-0 , p. 38.