Mathematical Space

A Mathematical Space is a mathematical set that ...

• Context:
  • It can (typically) have structures such as operations, relations, or metrics that define the space's properties.
  • It can (often) be studied through various branches of mathematics like algebra, geometry, analysis, and probability.
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  • It can range from being highly abstract, like an infinite-dimensional Hilbert Space, to very concrete, like the familiar Euclidean Space.
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  • It can be utilized to model real-world phenomena in physics, engineering, computer science, and other fields.
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• Example(s):
  • Basic Mathematical Spaces, such as:
    • Euclidean Space (ℝⁿ): The familiar n-dimensional real coordinate space.
    • Complex Space (ℂⁿ): The n-dimensional complex coordinate space.
    • Discrete Space: A space where every subset is open and closed.
  • Algebraic Spaces, such as:
    • Vector Space: A space closed under vector addition and scalar multiplication.
      • Examples: Coordinate spaces, function spaces, polynomial spaces.
    • Linear Space: Synonymous with vector space in most contexts.
    • Inner Product Space: A vector space with an inner product operation.
      • Example: Euclidean space with the dot product.
    • Module: A generalization of vector spaces over rings instead of fields.
  • Topological Spaces, such as:
    • Metric Space: A space with a distance function (metric) between points.
      • Examples: Euclidean space with Euclidean distance, function spaces with supremum norm.
    • Normed Vector Space: A vector space with a norm function.
      • Example: The space of continuous functions on [0,1] with the supremum norm.
    • Topological Vector Space: A vector space with a compatible topology.
    • Manifold: A space that locally resembles Euclidean space.
      • Examples: Sphere, torus, Möbius strip.
  • Functional Analysis Spaces, such as:
    • Banach Space: A complete normed vector space.
      • Example: The space of bounded continuous functions on [0,1] with supremum norm.
    • Hilbert Space: A complete inner product space.
      • Examples: L² spaces, sequence spaces like ℓ².
  • Measure and Probability Spaces, such as:
    • Measure Space: A space equipped with a measure function.
      • Example: Real line with Lebesgue measure.
    • Probability Space: A normalized measure space used in probability theory.
      • Example: A six-sided die with uniform probability distribution.
  • Geometric Spaces, such as:
    • Affine Space: A space that generalizes properties of Euclidean space.
    • Projective Space: A space of lines passing through the origin in a vector space.
      • Example: The real projective plane ℝP².
  • Advanced Spaces, such as:
    • Fock Space: Used in quantum mechanics for systems with varying particle numbers.
    • Sobolev Space: Function spaces with weak derivatives, important in PDEs.
    • Frechet Space: A locally convex topological vector space with a complete metric.
  • ...
• Counter-Example(s):
  • Discrete Sets without any additional structure, where no further mathematical properties are defined.
  • Arbitrary Collection of Points without a defined relationship, structure, or operations.
• See: Norm (Mathematics), Universe (Mathematics), Mathematical Structure.

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Space_(mathematics) Retrieved:2024-9-2.
  • In mathematics, a space is a set (sometimes known as a universe) endowed with a structure defining the relationships among the elements of the set.

    A subspace is a subset of the parent space which retains the same structure.

    While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can represent numbers, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms,and all three-dimensional Euclidean spaces are considered identical.

    Topological notions such as continuity have natural definitions for every Euclidean space. However, topology does not distinguish straight lines from curved lines, and the relation between Euclidean and topological spaces is thus "forgetful". Relations of this kind are treated in more detail in the "Types of spaces" section.

    It is not always clear whether a given mathematical object should be considered as a geometric "space", or an algebraic "structure". A general definition of "structure", proposed by Bourbaki, embraces all common types of spaces, provides a general definition of isomorphism, and justifies the transfer of properties between isomorphic structures.

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/space_(mathematics) Retrieved:2015-7-25.
  • In mathematics, a space is a set (sometimes called a universe) with some added structure.

    Mathematical spaces often form a hierarchy, i.e., one space may inherit all the characteristics of a parent space. For instance, all inner product spaces are also normed vector spaces, because the inner product induces a norm on the inner product space such that: : [math]\displaystyle{ \|x\| =\sqrt{\langle x, x\rangle}. }[/math] Modern mathematics treats "space" quite differently compared to classical mathematics.

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