nLab dual vector space

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 $K$ be a field (or any commutative rig), and let $V$ be a vector space (or module) over $K$.

Definition

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

More generally, let $K$ and $L$ be rings (or rigs) not assumed commutative, and let $V$ be a $K$-$L$-bimodule.

Definitions

The left dual module of $V$ is the right $K$-module $^*V$ of left $K$-module homomorphisms from $V$ to $K$. The right dual module of $V$ is the left $L$-module $V^*$ of right $L$-module homomorphisms from $V$ to $L$.

Now let $V$ be a topological vector space over the ground field $K$.

Definitions

(linear dual of a topological vector space)

Let $V$ be a topological vector space over the ground field $K$.

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

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 $K$ is an algebra over another rig $L$, then any $K$-module $V$ is also an $L$-module, but the dual as a $K$-module is different from the dual as an $L$-module. So one may speak of the $K$-dual or the dual over $K$.

• A topological vector space $V$ has an underlying discrete vector space, and these have different duals. So one speaks of the topological dual and the algebraic dual (respectively). If $V$ 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 $V \mapsto V^*$ extends to a contravariant functor.

Definition (transpose map)

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

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

for all $w$ in $W^*$ and $v$ in $V$.

Remark

This functor is, of course, the representable functor represented by $K$ 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 $B$ is any basis of $V$, then we can sometimes turn $B$ into a basis $B^*$ of the dual space $V^*$.

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

$$b^i(b_j) = \delta^i_j$$

(the Kronecker delta).

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

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

Double duals

The dual $\big(V^{\ast}\big)^\ast$ of the dual $V^\ast$ of a vector space $V$ is also called its double dual.

There is a natural transformation from $V$ to its double dual:

(In fact, historically this was the motivating example for the notion of natural transformations in the first place, see there)

The space $V$ is called reflexive if this natural transformation is an isomorphism. The reflexive spaces include all finite-dimensional vector spaces (or more generally modules) over fields (or more generall division rings), as well as all Hilbert spaces, the Lebesgue spaces $L^p$ over a localisable measure space for $1 \lt p \lt \infty$, 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 $V$ and $V^*$ does not make $Vect$ 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.

Examples

Example

(duality between vectors and covectors)

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

Example

(von Neumann algebras)

A von Neumann algebra (abstractly) is precisely a $C^*$-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 $C^*$-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).

Example

(spaces of distributions)

The standard topology on the spaces $\mathcal{D}'$ of distributions is the dual space topology according to def. .

(e.g. Hörmander 90, p. 38)

See Riesz representation theorem for more examples from functional analysis.

Related concepts

• dual space

• anti-dual linear space

• dual vector bundle

• bra-ket

• hyperbolic functor

References

Discussion in the context of distributions:

• Lars Hörmander, section 2.1 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990