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In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. Refer to the ball article for a more general introduction. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ...
A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...
1 (one) is the natural number following 0 and preceding 2. ...
A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...
Unit balls in normed vector spaces
More precisely, the open unit ball in a normed vector space V, with the norm , is In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
The word norm coming from the latin word norma which means angle measure or (lawlike) rule, has a number of meanings: A social or sociological norm; see norm (sociology). ...
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It is the interior of the closed unit ball of (V,||·||), In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...
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The latter is the disjoint union of the former and their common border, the unit sphere of (V,||·||), - .
Comments The 'shape' of the unit ball is entirely dependent on the chosen norm; it may well have 'corners', and for example may look like [−1,1]n, in the case of the norm l∞ in Rn. The round ball is understood as the usual Hilbert space norm, based in the finite dimensional case on the Euclidean distance; its boundary is what is usually meant by the unit sphere. In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...
Generalization to metric spaces All three of the above definitions can be straightforwardly generalized to a metric space, with respect to a chosen origin. However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces. In mathematics, a metric space is a set (or space) where a distance between points is defined. ...
In mathematics, an ultrametric space is a special kind of metric space. ...
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