Mathematical Space

Respond to this Friday Faithfuls challenge by writing anything about Math or Science, or you can go with whatever else that you think fits.  Math is not just about numbers, as it is the scientific study of motion, shape, space, connections and relationships, attributes and patterns that work with mathematical symbols.  Space is an abstract idea in math, it is extraordinarily general, but it is also a supremely important concept.  Space is where things happen, and spatial coordinates provide us with our reference system, they are necessary to define unique locations of any particular point or any object.  The words space and structure can be used in essentially all the same situations, but you could think of a space as being a geometric formation and a structure as more an algebraic foundation.  So, structures are places when we do algebra, and spaces as places when we do geometry.  Mathematical space makes me think about different shapes like those used in Geometry, and these objects are often defined as the mathematics of space.  Space gives us the geometry terms of distance or length, area and volume and this is sometimes called 1-dimensional, 2-dimensional or 3-dimensional space, however in math we will often work in higher dimensional spaces.  Knowing about these concepts can help us to understand complex phenomena.

Space can be looked at as being a place for numerical variables, data structures or functions to exist and space can comprise points on a line, objects in a plane or space can be in some higher dimensions of a universe.  Mathematics involves a certain sort of logical structure, requiring a logical mind to see how things fit together, as you can’t just go making things up.  This structure can be specified by a number of operations on the objects of the set.  A structure in math is the arrangement of and relations between the parts or elements of something complex.  These operations must satisfy certain general rules, called the axioms of the mathematical space.

Space is a geometric structure, having its own mathematical entities and axioms that define it.  Space is like a set (sometimes called a universe) with some added structure which could mean a variety of things, but typically it involves interactions and relationships between elements of the space, as well as rules on how to create and define new elements of the space.  The word space is used to describe individual examples of these abstract concepts and it is used throughout mathematics to refer to entirely different concepts in several distinct and unrelated contexts.  It carries no particular meaning, as it only loosely suggests a set with some structure and it is almost like a mathematical version of the word thing.  Mathematical space can be thought of as a creation which is used to work out specific problems.  Many types of spaces are named after people who were significant in developing their structure.

Various kinds of spaces exist such as affine space, ambient space, analytic space, Banach space, Besov space, Bochner space, Boolean space, Cartesian space, compact space, connected space, configuration space, coordinate space, Dirichlet space, dual space, Euclidean space, event space, feature space, Fock space, Fréchet space, function space, Fourier Space, Hausdorff space, Hardy space, hexaxial space, Hilbert space, Hölder space, hyperbolic space, inner-product space, isomorphic space, linear space, LF-space, L^p space, measure space, metric space, Minkowski space, moduli space, momentum space, Montel space, Morrey–Campanato space, Noncommutative space, normed space, null space, Orlicz space, outcome space, phase space, Poisson space, probability space, projective space, quotient space, Riemannian space, Riesz space, sample space, Schwartz space, Sobolev space, Stone space, subspace, tangent space, topological space, Tsirelson space, Unitary space, vector space and there is also a mathematical space called Swiss cheese which is an approximation that as you may have guessed is full of holes.

The most basic form of a space is a point and points can be combined or build up into a string of points which is where they eventually become lines, as lines are basically a series of points.  Space begins with the point.  When points are repeated, one after another in one direction, we go from a zero-dimensional object into the first dimension, and this new object is called a line.  An example of a one-dimensional space is the number line, where the position of each point on it can be described by a single number.

In geometry, Euclidean space is named after the Greek mathematician Euclid.  Its basic elements are points, out of which are constructed lines, planes, circles, triangles, etc. as sets of points having prescribed properties.  Euclidean space comes in dimensionalities 1, 2, and 3 (and higher if you like), denoted by common convention 𝔼1, 𝔼2, and 𝔼3 using the Double-Struck Capital letter.  Euclidean spaces are the simplest vector spaces, and an affine space is the same thing as a vector space, but there is no origin, so there is no way to distinguish where a point is located.  Euclidean geometry seeks to understand the geometry of flat, two-dimensional spaces, non-Euclidean geometry studies curved, rather than flat, surfaces.

Vectors in a Euclidean space form a linear space, but each vector has also a length and Euclidean space can be of any dimension, and it uses coordinates.  A vector space is a structure composed of vectors that has no magnitude or dimension.  Vectors represent a direction and magnitude where the direction can be specified by an angle and magnitude represents size, telling us how big something is, but vector spaces are just sets of objects where we can talk about adding the objects together and multiplying the objects by numbers.  A vector space or a linear space is a group of objects called vectors, added collectively and multiplied “scaled” by numbers, called scalars.  Scalars are usually considered to be real numbers.  Classical analytic geometry is the study of vector spaces over the real numbers.  These allow one to have a notion of direction, as well as the common understanding of “length”.  Metric spaces are the most general setting in which the concept of length makes sense.  A metric space is a set together with a notion of distance between its elements, that is usually called points.  The distance is measured by a function called a metric or distance function.

Cartesian Space was invented by René Descartes to create a coordinate system that could be used to define a point in space and it also comes in dimensionalities 1, 2, and 3 (and higher), denoted by ℝ1, ℝ2, and ℝ3.  A Cartesian space is a finite Cartesian product of the real line ℝ with itself.  The coordinates of a point on a cartesian plane are expressed as an ordered pair.  These points are signed and are located at a fixed distance from two perpendicular lines known as axes.  Cartesian coordinates allow one to specify the location of a point in the plane, or in three-dimensional space, and they can be used in three-dimensional space, based on three mutually perpendicular coordinate axes. x-axis, y-axis, and the z-axis, where the x and y denote width and height and the z denotes depth.  Coordinate systems are tools that let us use algebraic methods to understand geometry.  The Greeks used the concepts of angle and radius, but their mathematics did not extend to a full coordinate system.

Mathematical space can get very complicated, but I still want to discuss one of the more interesting spaces.  For quantum theory, space is considerably less arbitrarily defined, so this is where Hilbert space is used.  Heisenberg’s Uncertainty Principle states that we can’t know both the position and speed of a particle at the same time, but Hilbert space can be used to describe the possible states of the particles.  In order to understand Hilbert space, you will have a long road to travel, because this is very complex, but by knowing something about this, it can provide you with the mathematical foundations of quantum mechanics, thus giving you a method to think about quantum mechanical systems in general terms engaging your everyday geometric imagination.  I don’t want to confuse anyone by mentioning a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space, but I think you need to know about the term phase space, which refers to coordinates of a dynamic system in which the state of the system is recorded over time.  The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.  It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.  In quantum physics, the Hilbert space is the mathematical representation of all the possible states the system can take.

If you scream in Hilbert space, nobody will hear you!  When you scream, you emit sound waves, but at the quantum scale scientists will be unable to measure an energy that is ten septillion times smaller than the energy required to keep a lightbulb on for one second, unless they are using a quantum microphone, which is capable of picking up phonons which are packets of vibrational energy emitted by jittery atoms and are analogous to photons.