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The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. 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.
The association of a dual basis with a basis gives a map from the space of bases of V to the space of bases of V ∗, and this is also an isomorphism. For topological fields such as the real numbers, the space of duals is a topological space , and this gives a homeomorphism between the Stiefel manifolds of bases of these spaces.
In mathematics, a dual system, dual pair or a duality over a field is a triple (,,) consisting of two vector spaces, and , over and a non-degenerate bilinear map:.. In mathematics, duality is the study of dual systems and is important in functional analysis.
For example, the dual cone of the dual cone of a set contains the primal set (it is the smallest cone containing the primal set), and is equal if and only if the primal set is a cone. An important case is for vector spaces, where there is a map from the primal space to the double dual, V → V **, known as the "canonical evaluation map". For ...
The space of distributions, being defined as the continuous dual space of (), is then endowed with the (non-metrizable) strong dual topology induced by () and the canonical LF-topology (this topology is a generalization of the usual operator norm induced topology that is placed on the continuous dual spaces of normed spaces).
If is a Banach space, then so is its (continuous) dual. To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual space. In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, but in infinite dimensions the continuous ...
For example, if is the Banach space consisting of bounded functions on the real line with the supremum norm, then the map is not surjective. (See L p {\displaystyle L^{p}} space ). If φ {\displaystyle \varphi } is surjective, then X {\displaystyle X} is said to be a reflexive Banach space .
If g is a Lie algebra and π is a representation of it on the vector space V, then the dual representation π* is defined over the dual vector space V* as follows: [3] π*(X) = −π(X) T for all X ∈ g. The motivation for this definition is that Lie algebra representation associated to the dual of a Lie group representation is computed by the ...