Dual space

together with the vector space structure of pointwise addition and scalar multiplication by constants.

and the corresponding duality pairing are introduced, as described below in § Bilinear products and dual spaces.

consists of the space of geometrical vectors in the plane, then the level curves of an element of

can be intuitively thought of as a particular family of parallel lines covering the plane.

is a vector space of any dimension, then the level sets of a linear functional in

, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.

, then the same construction as in the finite-dimensional case yields linearly independent elements

[10] Any bilinear form ⟨·,·⟩ on V gives a mapping of V into its dual space via where the right hand side is defined as the functional on V taking each w ∈ V to ⟨v, w⟩.

from V to a subspace of V∗ (resp., all of V∗ if V is finite dimensional) defines a unique nondegenerate bilinear form

can be identified with the set of all additive complex-valued functionals f : V → C such that There is a natural homomorphism

This identity characterizes the transpose,[12] and is formally similar to the definition of the adjoint.

Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the orthogonal complement.

with its image in the second dual space under the double duality isomorphism

In particular, forming the annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space.

to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to reducing a fraction.

This arises in physics via dimensional analysis, where the dual space has inverse units.

[13] Under the natural pairing, these units cancel, and the resulting scalar value is dimensionless, as expected.

This is however false for any infinite-dimensional normed space, as shown by the example of discontinuous linear maps.

is supposed to satisfy the following conditions: If these requirements are fulfilled then the corresponding topology on

Conversely, given an element a = (an) ∈ ℓ q, the corresponding continuous linear functional

This gives rise to the bra–ket notation used by physicists in the mathematical formulation of quantum mechanics.

For example, the bounded linear map T has dense range if and only if the transpose T′ is injective.

When V is a Hilbert space, there is an antilinear isomorphism iV from V onto its continuous dual V′.

[21] Indeed, let P denote the canonical surjection from V onto the quotient V / W ; then, the transpose P′ is an isometric isomorphism from (V / W )′ into V′, with range equal to W⊥.

If j denotes the injection map from W into V, then the kernel of the transpose j′ is the annihilator of W: and it follows from the Hahn–Banach theorem that j′ induces an isometric isomorphism

In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Ψ : V → V′′ from a normed space V into its continuous double dual V′′, defined by As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning ‖ Ψ(x) ‖ = ‖ x ‖ for all x ∈ V. Normed spaces for which the map Ψ is a bijection are called reflexive.

When V is a topological vector space then Ψ(x) can still be defined by the same formula, for every x ∈ V, however several difficulties arise.

First, when V is not locally convex, the continuous dual may be equal to { 0 } and the map Ψ trivial.

[nb 5] Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual V′, so that the continuous double dual V′′ is not uniquely defined as a set.

As a consequence, defining reflexivity in this framework is more involved than in the normed case.