A pair of dual vector spaces, is a particular case of duality system.

It is a pair of sets (

They are formally written as follows:

∀u,v∈E, (∀x∈E', u⋅x = v⋅x) ⇒ u=v

∀x,y∈E', (∀u∈E, u⋅x = u⋅y) ⇒ x=y

In more intuitive words : Each element of *E* is completely
specified by its scalar products by the elements of *E'* ;
and the same exchanging *E* and *E*'.

In more sophisticated words : both curried forms of the scalar
product : a map from E to ℝ^{E'}, and a map from *E
*' to ℝ^{E}, are
injective; thus *E* plays the role of (is identifiable with)
a subset of ℝ^{E '},
and *E*' plays the role of a subset of ℝ^{E
'}as identified to its image in ℝ* ^{E}*.

We have a zero element in E, whose scalar product with any
element of E' is zero; and a zero element of E', whose scalar
product by any element of E is zero. We shall abusively denote the
zeros of all sets by the same symbol 0, letting their precise
identity be given by the operations where they are used, as this
won't cause any ambiguity:

- ∀x∈E', 0⋅x = 0
- ∀u∈E', u⋅0 = 0

We define an operation of addition in E as the addition of
the scalar products with any element of E' ; and the same in E'.
Formally:

- ∀u,v∈E, ∀x∈E', (u+v)⋅x = u⋅x + v⋅x
- ∀x,y∈E', ∀u∈E, u⋅(x+y) = u⋅x + u⋅y

In other words : for all u,v∈E, the function from E' to R defined by (x ↦ u⋅x + v⋅x) "belongs to" (or : is represented through the scalar product by) an element of E that is denoted u+v.

We define an operation of multiplication of elements of E by any
real number, as multiplying by the same number, the scalar product
with any element of E'. And the same in E'. Formally :

- ∀a∈ℝ, ∀u∈E, ∀x∈E', (au)⋅x = a(u⋅x) = u⋅(ax)

Remark : the zero element of E can as well be obtained from any
element u of E by the multiplication by 0∈ℝ: 0 = 0u. But declaring
the constant 0∈E further says that E is nonempty.

The above definitions of the operations in E can also be
equivalently expressed in the following two ways

- E represents (though the scalar product) a vector subspace of the space of functions from E' to ℝ. (stable by addition and multiplication by a scalar).
- E is a vector space, and the elements of E' represent linear functions from E to ℝ.

(the below will be reworked later)

If (E,E') is a pair of dual spaces and E is finite-dimensional then E' is the space of all linear forms on E and has the same dimension.

There are counter-examples in the infinite-dimensional case. For example take E = E' = the set of continuous maps from [-1,1] to ℝ, and the operation of integral of the product of these functions.

Then the Dirac mass in 0 (or in any other point of [0,1], which maps any f in E to f(0), is outside E'. It may still be understood as a limit of a sequence of elements of E'.

(This page will be later expanded to further details)

Note : instead of ℝ we may as well take ℂ or generally any field. Eventually some other commutative ring may be used but this can make some of the below constructions fail)

Notice that the set of "rotations" (transformations which preserve the inner product) has dimension n(n-1)/2 for any signature.

Next:

- List of physics theories
- Affine geometry (to be developed later)
- Special relativity
theory

- Introduction to tensors