Dual Space in a sentence

Nevertheless, in the theory of topological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space", since there is no serious need to consider discontinuous maps in this field.

There are two types of dual spaces: the algebraic dual space, and the continuous dual space.

Example: dual of a finite-dimensional vector space Every finite-dimensional vector space is isomorphic to its dual space, but there may be many different isomorphisms between the two spaces.

For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general.

When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.

A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space.

This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it.

In Banach spaces, a large part of the study involves the dual space : the space of all continuous linear maps from the space into its underlying field, so-called functionals.

Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the dual space above.

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.

In that approach a type (p, q) tensor T is defined as a map, : where V is a (finite-dimensional) vector space and V* is the corresponding dual space of covectors, which is linear in each of its arguments.

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X*, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak-* topology.

The coefficients that bound the inequalities in the primal space are used to compute the objective in the dual space, input quantities in this example.

The coefficients used to compute the objective in the primal space bound the inequalities in the dual space, output unit prices in this example.

The space of distributions on U is denoted by D′(U) and it is the continuous dual space of D(U).

This allows us to use the Riesz representation theorem and find that the dual space of ℓ ∞ (N) can be identified with the space of finite Borel measures on βN.

This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional.

Thus if the basis is infinite, then the algebraic dual space is always of larger dimension (as a cardinal number ) than the original vector space.

To each inequality to satisfy in the primal space corresponds a variable in the dual space, both indexed by input type.

In the dual space, it expresses the creation of the economic values associated with the outputs from set input unit prices.