Given any vector space V over some field F, we define the dual space V* to be the set of all linear functions from V to F. V* itself becomes a vector space over F if addition and scalar multiplication are defined as follows:
If the dimension of V is finite, then V* has the same dimension as V; if {ei} is a basis for V, then the associated dual basis {ei} of V* is given by
| 1, if i = j ei(ej) = | | 0, if i ≠ j.
If V is infinite-dimensional, however, then the above construction does not produce a basis for V* and the dimension of V* is greater than that of V.
If f: V -> W is a linear map, we may define its transpose tf : W* -> V* by
As we saw above, if V is finite-dimensional, then V is isomorphic to V*, but the isomorphism is not natural and depends on the basis of V we started out with. In fact, any isomorphism Φ from V to V* defines a unique [bilinear product]? on V by
<v,w> = (Φ(v))(w)
and vice versa.
There is a natural homomorphism Ψ from V into the double dual V**, defined by (Ψ(v))(φ) = φ(v) for all v in V, φ in V*. This is always injective and, in the event that V is [finite dimensional]?, is actually an isomorphism.
When dealing with a normed vector spaces V such as a Banach spaces or a Hilbert spaces, one typically is only interested in the continuous linear functionals from the space into the base field. These form a normed vector space, called the continuous dual of V, sometimes just called the dual of V. The norm ||φ|| of a continuous linear functional on V is defined by
In analogy to the case of the algebraic double dual, there is always a naturally defined injective continuous linear operator from V into its continous double dual. Spaces for which this map is a bijection are called reflexive?.