Metric Space (Mathematics)

Metric space is a mathematical way of defining distances and lengths between known elements within a set. A set is defined as any collection of distinct objects that, once brought together, may be viewed as a single, collective object. Sets may be composed of any number of different types of things, ranging from people or buildings to more abstract mathematical components. Each member of a set is called an element of the set. For instance, a city may be viewed as a type of set. Each building within the city may be then considered as an element of the set. The term metric is defined as the distance between each pair of elements within a set (or, in the example above, between two separate buildings in the city), while a metric space is a set in which all the distances between each pair of elements has been defined. Calculations using metric space allow these distances to be assigned a positive real number (which is any number greater than zero) for every pair of elements.

Background

Metric space was first defined by French mathematician Maurice Fréchet. Fréchet was born the fourth of sixth children to Jacques and Zoé Fréchet. His father was an educator who was the director of an orphanage at the time of his son's birth before eventually becoming the head of a Protestant school. Fréchet studied under Jacques Hadamard, who was an expert in number theory. Hadamard identified Fréchet's mathematical talents and personally tutored him, eventually supervising the composition of Fréchet's doctoral dissertation.

This dissertation provided the origins for metric space. Submitted in 1906, it was entitled Sur quelques points du calcul fonctionnel (On Some Points of Functional Calculus) and helped to establish the underlying frameworks for this new field of mathematical study. Fréchet also made important contributions to the study of other abstract spaces such as topological space and vector space.

Fréchet served variously as a professor of calculus, mechanics, and mathematics over the course of his academic career. He is regarded as a pioneer in the foundation of abstract space theory who helped to propose new arenas of study. He also made important contributions that helped to advance such mathematical fields as topology, statistics, and probability.

Overview

Metric space is useful for determining distances in mathematical analyses. It has particularly valuable applications in the examinations of abstract topologies. Topology is the study of different types of space that are able to remain flexible without breaking. Topology examines the ways in which any space (such as a triangle) can be stretched or pulled without being torn. Therefore, under the topological definition of space, there is little practical difference between circles, triangles, and squares because all may be twisted to eventually form the same shape. However, once any shape becomes fixed into a specific form in which it will be broken if stretched, it then is defined under the terms of Euclidean space. For instance, topological spaces may be seen as a type of shape made of a soft flexible substance like clay. Clay may be pulled or bent to form a number of different shapes, such as a flat circle. The clay circle can even be easily reformed into a three-dimensional rounded bowl. However, once the clay has been fired in a kiln, it is now fixed permanently into a specific form, such as a solid bowl that cannot be reshaped without being broken. The malleable clay is a topologic space, while the fired bowl is a Euclidean space.

To cite another example using the example of a city, a Euclidean distance would be a direct line moving from start to finish in a straight, fixed direction. However, in the real world, it is often impossible to move in a perfectly straight line, as objects and buildings would be in the way. Instead, movement through the city would require various (flexible) movements left and right. Figuring out this more abstract distance is a measurement of metric space. This principle is called the taxicab metric due to its similarity with a network of city roads when graphed.

To be measured as metric space, the distance between elements must satisfy certain mathematical conditions. First, the distance (defined as d) between the first point of a measured set (point X) and itself (also point X) is zero. This function enables the measurement of any distances to begin at a fixed, non-negative point. Second, the distance between the first point in the set (point X) and any second point (point Y) is equal to the distance between the second point (point Y) and the first point (point X). This means that the distance between these two points remains symmetrical and constant, and does not change if measured from X to Y or from Y to X. Finally, for any three points (points X, Y, and Z), the measured distance between points X and Z cannot be more than the combined distance between points X and Y when added to the distance between points Y and Z. A real-world way of thinking about this is to use the example of the city above. If someone were to travel in the city from point X to Y and then from point Y to Z, this condition requires that any extra stops could not make the trip shorter. However, any such extra stops are capable of making the trip longer. This last principle is intended to satisfy the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the measurement of the third side.

There are different types of mathematical space that are defined by strict sets of unique characteristics. These include inner product space, normed vector space, metric space, and topological space. There is also a hierarchy of these types of space. This means that any type of space that ranks lower on the taxonomic scale contains the characteristics of the type of space placed above it. For instance, inner product space, which ranks lowest, is also a type of vector space and metric space, both of which are ranked sequentially higher. However, not all metric space can be classified as either inner product space or vector space. To give an example of this manner of categorization, all apples are a type of fruit, but not all fruits are a type of apple. Therefore, all apples have the characteristics of fruit, but fruit as a whole may have individual traits not found in apples. However, all of these examples of space are regarded as types of abstract space.

Bibliography

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