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Reflexive space

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Bornological space

In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

Dual norm

In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Let be a normed vector space with norm and let denote its continuous dual space. The dual norm of a continuous linear functional belonging to is the non-negative real number defined by any of the following equivalent formulas: where and denote the supremum and infimum, respectively.

Semi-reflexive space

In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is bijective. If this map is also an isomorphism of TVSs then it is called reflexive. Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable.

Topologies on spaces of linear maps

In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The article operator topologies discusses topologies on spaces of linear maps between normed spaces, whereas this article discusses topologies on such spaces in the more general setting of topological vector spaces (TVSs).

Schwartz space

In mathematics, Schwartz space is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of , that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French mathematician Laurent Schwartz.

Weak topology

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.

Dual space

In mathematics, any vector space has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.

Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V. Let be the vector space equipped with the usual dot product (thus making it an inner product space), and let with then its orthogonal complement can also be defined as being The fact that every column vector in is orthogonal to every column vector in can be checked by direct computation.

Schauder basis

In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described by Juliusz Schauder in 1927, although such bases were discussed earlier.

Uniformly convex space

In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. A uniformly convex space is a normed vector space such that, for every there is some such that for any two vectors with and the condition implies that: Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short.