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Probabilistic metric spaces are initially introduced by Menger, which were termed statistical metrics. [3] Shortly after, Wald criticized the generalized triangle inequality and proposed an alternative one. [ 4 ]
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets.
Uniform Boundedness Principle — Let be a Banach space, a normed vector space and (,) the space of all continuous linear operators from into . Suppose that F {\displaystyle F} is a collection of continuous linear operators from X {\displaystyle X} to Y . {\displaystyle Y.}
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set () of -dimensional normed spaces. With this distance, the set of isometry classes of n {\displaystyle n} -dimensional normed spaces becomes a compact metric space , called the Banach–Mazur compactum .
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. An unusual property of normed vector spaces is that linear transformations between them are continuous if and only if ...
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]
If Y is a Banach space, an equivalent definition is that the embedding operator (the identity) i : X → Y is a compact operator. When applied to functional analysis, this version of compact embedding is usually used with Banach spaces of functions. Several of the Sobolev embedding theorems are compact embedding theorems.
Inner product spaces are a subset of normed vector spaces, which are a subset of metric spaces, which in turn are a subset of topological spaces. In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. [1]