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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. These linear functions to the base field are also called linear functionals. V* itself becomes a vector space over F under the following definition of addition and scalar multiplication:
If the dimension of V is finite, then V* has the same dimension as V; if {e_{1},...,e_{n}} is a basis for V, then the associated dual basis {e^{1},...,e^{n}} of V* is given by
Concretely, if we intepret R^{n} as space of columns of n real numbers, its dual space is typically written as the space of rows of n real numbers. Such a row acts on R^{n} as a linear functional by ordinary matrix multiplication.
If V consists of the space of geometrical vectors (arrows) in the plane, then the elements of the dual V* can be intuitively represented as collections of parallel lines. Such a collection of lines can be applied to a vector to yield a number in the following way: one counts how many of the lines the vector crosses.
If V is infinitedimensional, then the above construction of e^{i} does not produce a basis for V* and the dimension of V* is greater than that of V. Consider for instance the space R^{(ω)}, whose elements are those sequences of real numbers which have only finitely many nonzero entries. The dual of this space is R^{ω}, the space of all sequences of real numbers. Such a sequence (a_{n}) is applied to an element (x_{n}) of R^{(ω)} to give the number ∑_{n}a_{n}x_{n}.
As we saw above, if V is finitedimensional, 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 nondegenerate bilinear product[?] on V by
and conversely every such nondegenerate bilinear product on a finitedimensional space gives rise to an isomorphism from V to V*.
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 map Ψ is always injective; it is an isomorphism if and only if V is finite dimensional[?].
When dealing with a normed vector space V (e.g., a Banach space or a Hilbert space), 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. It is denoted by V '. The norm φ of a continuous linear functional on V is defined by
One may also talk about the continuous dual of an arbitrary topological vector space. This is however much harder to deal with since it will in general not be a normed vector space in any natural way.
For any finitedimensional normed vector space or topological vector space, such as Euclidean nspace, the continuous dual and the algebraic dual coincide.
Let 1 < p < ∞ be a real number and consider the Banach space l^{ p} of all sequences space of sequences a = (a_{n}) for which
In a similar manner, the continuous dual of l^{ 1} is naturally identified with l^{ ∞}. Furthermore, the continuous duals of the Banach spaces c (consisting of all convergent sequences, with the supremums norm) and c_{0} (the sequences converging to zero) are both naturally identified with l^{ 1}.
If V is a Hilbert space, then its continuous dual is a Hilbert space which is antiisomorphic to V. This is the content of the Riesz representation theorem, and gives rise to the braket notation used by physicists in the mathematical formulation of quantum mechanics.
In analogy with the case of the algebraic double dual, there is always a naturally defined injective continuous linear operator Ψ : V → V '' from V into its continuous double dual V ''. This map is in fact an isometry, meaning Ψ(x) = x for all x in V. Spaces for which the map Ψ is a bijection are called reflexive.
The continuous dual can be used to define a new topology on V, called the weak topology.
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