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Types of spaces. Linear and topological spaces. Affine and projective spaces. Metric and uniform spaces. Normed, Banach, inner product, and Hilbert spaces. Smooth and Riemannian manifolds. Measurable, measure, and probability spaces. Non-commutative geometry. Schemes. Topoi. Spaces and structure. List of mathematical spaces. See also. Notes.
Two basic spaces are linear spaces (also called vector spaces) and topological spaces. Linear spaces are of algebraic nature; there are real lin-ear spaces (over the field of real numbers), complex lin-ear spaces (over the field of complex numbers), and more generally, linear spaces over any field.
A metric space is made up of a nonempty set and a metric on the set. The term “metric space” is frequently denoted (X, p). The triangle inequality for the metric is defined by property (iv). The set R of all real numbers with p(x, y) = | x – y | is the classic example of a metric space. Introduction to Metric Spaces
Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs.
May 3, 2023 · Since metric spaces are not ordered, concepts and results concerning the real numbers that depend on order for their definitions must be redefined and reexamined in the context of metric spaces. The first example of this kind is completeness.
Aug 28, 2014 · The general concept of a mathematical space as a "multiple extension", or "manifold", was introduced in 1854 by B. Riemann. It was generalized, axiomatized and made concrete in various directions: such as, e.g., Riemannian space; Finsler space; vector space; Hilbert space; metric space, and topological space.
May 15, 2024 · Affine Space, Baire Space, Banach Space, Base Space, Bergman Space, Besov Space, Borel Space, Calabi-Yau Space, Cellular Space, Chu Space, Dimension, Drinfeld's Symmetric Space, Eilenberg-Mac Lane Space , Euclidean Space, Fiber Space, Finsler Space, First-Countable Space, Fréchet Space, Function Space, G-Space, Green Space, Heisenberg S...
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