Search results
Results from the WOW.Com Content Network
If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by . A norm induces a distance, called its (norm) induced metric, by the formula (,) = ‖ ‖. which makes any normed vector space into a metric space and a topological vector space.
A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". [1]
The Lebesgue space. The normed vector space ((,), ‖ ‖) is called space or the Lebesgue space of -th power integrable functions and it is a Banach space for every (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem).
More generally, the Kuratowski embedding allows one to see any metric space as a subspace of a normed vector space. Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in functional analysis. Completeness is particularly important in this context: a complete normed vector space is known as a Banach space.
The Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance. It is named after the Polish mathematician Hermann Minkowski .
When X is a vector space and the two metrics and are those induced by norms ‖ ‖ and ‖ ‖, respectively, then strong equivalence is equivalent to the condition that, for all , ‖ ‖ ‖ ‖ ‖ ‖ For linear operators between normed vector spaces, Lipschitz continuity is equivalent to continuity—an operator satisfying either of these ...
Fig. 1: Overview of types of abstract spaces. An arrow indicates is also a kind of; for instance, a normed vector space is also a metric space. 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 ...
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space.Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.