Vector Spaces in a sentence

Direct product and direct sum main The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.

Vector spaces with additional structure From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension.

Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.

The dual space of a finite-dimensional vector space is again a finite-dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the only invariant of finite-dimensional vector spaces over a given field.

All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently.

In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.

The elements of topological vector spaces are typically functions or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.

Their study—a key piece of functional analysis —focusses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.

Hermitian vector spaces and spinors If the vector space V has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural.

Linear transformations main Similarly as in the theory of other algebraic structures, linear algebra studies mappings between vector spaces that preserve the vector-space structure.

Vector spaces main Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.

When geometries are constructed from vector spaces, these two notions of dimension can lead to confusion, so it is often the case that the geometric concept is called geometric or projective dimension and the other is algebraic or vector space dimension.

Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.

Function spaces Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication pointwise.

More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.

Roughly, affine spaces are vector spaces whose origins are not specified.

The only difference is that we call spaces like this V modules instead of vector spaces.

The superstructure over the real numbers includes a wealth of mathematical structures: For instance, it contains isomorphic copies of all separable metric spaces and metrizable topological vector spaces.

This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

Actions of groups on vector spaces are called representations of the group.