If a distance concept doesn’t exist, a continuum concept can’t exist. That is similar to the familiar distance from three dimensional space, then a continuum ofĭistances will exist. That implies a continuum of possible distances. ![]() Well, the amount of separation can be anything from infinitely small In the above dictionary definition distance is defined as the extent of spatial And closelyĬonnected with the ideas of distance and a continuum is the idea of adjacency, being “infinitely The concept of distance is intricately tied to the concept of a continuum of points. That which is characterized by dimensions extending indefinitely in all directionsĪnd within which all material bodies are located. ![]() The extent of spatial separation between things, places or locations.ĭef. What is distance? We are all familiar with the idea of the distanceįirst let us look at the dictionary definition.ĭef. Let us give some thought to theĬoncept or idea of distance. This metric on a normed linear space is called the induced metric.Ĭoncepts: distance idea, continuum of points. Where x and y are vectors (or points) in the space and || x - y || is the norm of the vector x - y. For information on normed linear spaces see the following:Īny normed linear space can be turned into a metric space by defining on it the distance function Most of the spaces of major importance in analysis are normed linear The metric spaces of Examples 1 - 5 above areĪll normed linear spaces. Where P 1(x 1, y 1, z 1) and P 2(x 2, y 2, z 2) are any two points of the space. The set of all points (x, y, z) of three dimensionalĮuclidean space with a distance function defined by Where P 1(x 1, y 1) and P 2(x 2, y 2) are any two points of the space. Space with a distance function defined by The set of all points (x, y) of two dimensional Euclidean The set R of all real numbers in which the distance function is the usualĭistance between points on the real line given by d(x, y) = |x - y|. It is the set M along with a specified distance function (orġ. It often happens that several different metrics can beĭefined on a set of elements giving different metric spaces. One as in the case of a functional space. The distance function associated with a metric space may be a naturally occurring one as in theĬase of the usual Euclidean spaces of one, two or three dimensions or it may be a man-defined “points” x and y of a metric space is called a metric or distance function. The distance d(x, y) that is defined between The above properties correspond to certain central properties ofĭistances in three dimensional Euclidean space. Y so that the following conditions, namely the axioms of a metric space, are satisfied:Ī metric space is an abstract mathematical system, a generalization/ abstraction of threeĭimensional Euclidean space. a number d(x, y) is associated with each pair of points x, ![]() A metric space consists of a set M of arbitrary elements, called points,īetween which a distance is defined i.e. Convergence of sequences.Ĭauchy’s condition for convergence.
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