Dual space

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Algebraic Dual Space

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:

(φ + ψ)(x) = φ(x) + ψ(x)
(aφ) (x) = a φ(x)

for all φ, ψ in V*, a in F and x in V. In the language of tensors, elements of V are sometimes called contravariant vectors, and elements of V*, covariant vectors or one-forms.

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 ij.

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

tf (φ) = φ o f    for every φ in W*.

The assignment f |-> tf provides an injective homomorphism between the space of linear operators from V to W and the space of linear operators from W* to V*; this homomorphism is an isomorphism iff W is finite-dimensional. If the linear map f is represented by the matrix A with respect to two bases of V and W, then tf is represented by the transposed matrix tA with respect to the dual bases of W* and V*. If g: W -> X is another linear map, we have t(g o f) = tf o tg. In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself.

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.

Continuous Dual Space

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

||φ|| = sup { ||φ(x)|| : ||x|| ≤ 1 }

If V is a Banach space, then its continuous dual is also a Banach space; if V is a Hilbert space, then its continuous dual is a Hilbert space which is anti-isomorphic to V, the content of the Riesz representation theorem which gives rise to the bra-ket notation used by physicists in the mathematical formulation of quantum mechanics.

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.


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