nLab
dual vector space

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Homological algebra

homological algebra

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nonabelian homological algebra

Context

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Duality

Contents

Idea

A dual vector space is a dual in a closed category of vector spaces (or similar algebraic structures).

Of course, this is a very restricted notion of space; but for spaces in geometry, one usually uses the duality between space and quantity and speaks of the spectrum (not ‘dual space’) of an algebra. In homotopy theory, there are also Spanier-Whitehead duals; and there are other notions of dual space in physics.

Definitions

Let KK be a field (or any commutative rig), and let VV be a vector space (or module) over KK.

Definition

The dual space or dual module of VV is the vector space V *V^* of linear functionals on VV. That is, V *V^* is the internal hom [V,K][V,K] (thinking of KK as a vector space over itself: a line).

More generally, let KK and LL be rings (or rigs) not assumed commutative, and let VV be a KK-LL-bimodule.

Definitions

The left dual module of VV is the right KK-module *V^*V of left KK-module homomorphisms from VV to KK. The right dual module of VV is the left LL-module V *V^* of right LL-module homomorphisms from VV to LL.

Now let VV be a topological vector space over the ground field KK.

Definitions

The dual space of VV is the topological vector space V *V^* of continuous linear functionals on VV, equipped with the weak-* topology? (meaning the initial topology generated by the elements of VV, viewed as themselves linear functionals on V *V^*).

In principle, there is no conflict among these definitions, the most general case (so far) being a topological bimodule over two topological rigs; the non-topological cases simply involve discrete spaces. In practice, however, some complications are possible:

  • If the rig KK is an algebra over another rig LL, then any KK-module VV is also an LL-module, but the dual as a KK-module is different from the dual as an LL-module. So one may speak of the KK-dual or the dual over KK.

  • A topological vector space VV has an underlying discrete vector space, and these have different duals. So one speaks of the topological dual and the algebraic dual (respectively). If VV is considered with several different topologies (say called ‘weak’ and ‘strong’), then one may speak of the weak dual and the strong dual (etc).

Logically, these duals take place in different categories, which are related by various functors; the objects whose duals are being taken should not be conflated. In practice, however, these objects are identified, so the duals must be distinguished.

Transpose maps

The operation VV *V \mapsto V^* extends to a contravariant functor.

Definition (transpose map)

The dual linear map or transpose map of a linear map A:VWA\colon V\to W, is the linear map A *=A T:W *V *A^* = A^T\colon W^*\to V^*, given by

A *(w),v=w,A(v) \langle{A^*(w), v}\rangle = \langle{w, A(v)}\rangle

for all ww in W *W^* and vv in VV.

Remark

This functor is, of course, the representable functor represented by KK as a vector space over itself (a line).

Remark

This construction is the notion of dual morphism applied in the monoidal category Vect with its tensor product monoidal structure.

Dual bases

If BB is any basis of VV, then we can sometimes turn BB into a basis B *B^* of the dual space V *V^*.

We will begin with the definition of what might be the dual basis, cautioning that it is not always a basis:

Definition (dual basis)

Treating the basis BB as a family (b i)(b_i) with index set II, the dual basis B *B^* is the family (b i)(b^i) (with the same index set) such that

b i(b j)=δ j i b^i(b_j) = \delta^i_j

(the Kronecker delta).

Since BB is a basis of VV, this formula defines b ib^i for each index II, so B *B^* exists; but in general there is no reason why B *B^* should be a basis of V *V^*. However, if VV has finite dimension, then B *B^* is a basis of V *V^*. If VV is a Hilbert space, and BB is a basis of VV in the Hilbert space sense (i.e., BB is a linearly independent set whose span is topologically dense in VV), then also B *B^* is a basis of the dual Hilbert space V *V^*.

This is related to but different from the sort of dual basis applicable generally to projective modules.

Double duals

The double dual? of VV is simply the dual of the dual of VV. There is a natural transformation from VV to its double dual:

x^(λ)=λ(x), \hat{x}(\lambda) = \lambda(x) ,

for xx in VV, λ\lambda in V *V^*, and consequently x^\hat{x} in V **V^{**}.

The space VV is called reflexive if this natural transformation is an isomorphism. The reflexive spaces include all finite-dimensional vector spaces or modules over fields or division rings, as well as all Hilbert spaces, the Lebesgue spaces L pL^p over a localisable measure space for 1<p<1 \lt p \lt \infty, and others.

Dual spaces as dual objects

A dual vector space is a dual object in the monoidal category Vect equipped with its tensor product monoidal structure.

In general, the duality between VV and V *V^* does not make VectVect into a monoidal category with duals. However, if we restrict to spaces of finite dimension, then we get a compact category; finite-dimensional Hilbert spaces form a \dagger-compact category, which is very nice indeed.

Applications

A von Neumann algebra (abstractly) is precisely a C *C^*-algebra whose underlying Banach space is the dual space of some (other) Banach space. One may equivalently define a von Neumann algebra as a Banach space together with a C *C^*-algebra structure on its dual space (except that the morphisms go the other way, so one is more directly defining a noncommutative measurable space, along the lines of noncommutative geometry).

Tensors in finite-dimensional differential geometry make heavy use of the duality between tangent spaces and cotangent spaces. Infinite-dimensional differential geometry is much harder, largely because this duality is no longer perfect.

Examples

See Riesz representation theorem for examples from functional analysis.

Revised on January 3, 2014 10:57:10 by Toby Bartels (98.19.45.214)