Maximum norm

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The maximum norm , maximum norm or Chebyshev norm is a special norm for functions or for vectors or matrices . It is a special case of the supreme norm .

definition

Let be a compact space and the set of all real or complex valued continuous functions . Then the function is called through

is defined, maximum norm. The function is also referred to as and fulfills the three characteristic properties of a standard . The maximum norm is well defined on the basis of the principle of minimum and maximum , which ensures the existence of the maximum.

properties

  • The set of continuous functions on a compact set is a complete normalized space with the maximum norm .
  • Together with the product , the normalized space is a commutative Banach algebra .

Special cases

An important special case is the maximum norm for vectors . If you choose and equip the set with the discrete topology , then there is a compact space and every real or complex-valued function on is continuous . Thus the space corresponds to the -dimensional vector space and the maximum norm on vectors is a special case of the maximum norm for continuous functions on compact sets. If one sees a matrix as a correspondingly long vector im , it is also possible to define the maximum norm on matrices.

As a vector norm

For a vector one calls

the maximum norm of . The maximum norm can also be seen as a borderline case of the p norms . If we let it approach infinity, we get the maximum norm from the p norm. For this reason, the maximum norm for vectors is also referred to as the norm (infinity norm).

The spheres with respect to the maximum norm are precisely the -dimensional cubes , the edges of which all run parallel to the coordinate axes. The extreme points of such a closed “sphere” are precisely the corner points of this cube. The set of these points is (for ) a real subset of the edge of the cube, which consists of all edge (hyper) surfaces of the cube. with the maximum norm for is therefore a not strictly convex space . Nevertheless, the maximum norm is equivalent to the Euclidean norm , which makes it strictly convex.

As a matrix norm

Analogous to the vector norm, the maximum norm for matrices has the representation

However, this norm is not sub-multiplicative , which is why the overall sub-multiplicative norm is often used instead of this norm in connection with matrices .

Examples

Column vector

The following applies to the column vector

So the maximum norm of is 9.

function

For the fractional rational function defined by applies

This can be shown by two-fold derivation and determination of the extreme values . The maximum norm of the function on the interval is therefore 1.

Supreme norm

In contrast to the maximum norm, the supremum norm is not defined for continuous, but for limited functions . In this case it is not necessary that it is compact; can be any amount. Since continuous functions are restricted to compact spaces, the maximum norm is a special case of the supremum norm.

illustration

In clear terms, the distance derived from the maximum norm is always relevant if one can move in a multidimensional space in all dimensions simultaneously and independently of one another at the same speed.

The movement of a king on a chessboard can serve as a simple example of this : According to the rules, the king can move to an adjacent line or row in one move, whereby both can be combined (diagonal move). In order to determine the minimum number of moves a king needs to move from one square to another, one has to determine the maximum number of row changes and line changes to be made. If you represent a field by an ordered pair of the numbers 1, ..., 8, you need straight from field to field

Trains.

Example: The fields b8 and f3 of the chessboard are represented in this notation by the pairs and . A king therefore needs moves to move from one square to the other. It was not taken into account that the path can be blocked by your own or opposing pieces. In addition, the possibility of castling was ignored in the considerations .

More generally, the maximum norm can be used to determine how fast one can move in a two- or three-dimensional space, assuming that the movements in -, - (and -) direction occur independently, simultaneously, and at the same speed.

A system can be considered even more generally, the state of which is determined by independent parameters. Changes can be made to all parameters at the same time and without influencing one another. Then the maximum norm "measures" the time it takes to transfer the system from one state to another. A prerequisite for this, however, is that the parameters have been standardized in such a way that the same intervals between the values ​​also correspond to the same change times. Otherwise one would have to use a weighted version of the maximum norm that takes into account the different rates of change of the parameters.

Individual evidence

  1. Chebyshev norm . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .
  2. maximum norm . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .
  3. a b Alt: Linear functional analysis . 5th edition. Springer, 2006, ISBN 3-540-34187-0 , pp. 38 .
  4. a b Harro Heuser: Textbook of Analysis. Part 2. 14th edition Teubner Verlag, 2008, ISBN 978-3-8351-0208-8 , pp. 11-12.