Conceiving affine and vector spaces as ℝ-duality systems, reduces the issue of formalizing affine and linear geometries, to the following questions:

- Describing ℝ with its algebraic structures, and its axioms (including second-order ones);
- We shall accept all duality morphisms. Works of functional analysis involving more primitive structures to restrict the class of morphisms, are beyond the scope of this exposition.
- Specify the axioms (criteria to accept duality systems as spaces):
*E*and*E'*will be made algebras by stability axioms*E*∈ Sub ℝ^{E'}and*E'*∈ Sub ℝ^{E}, from different algebraic structures on ℝ which we have to describe. That will be all for us.

- A
*vector space*is an affine space*E*with a chosen point 0_{E}called "origin" - A
*linear map*between vector spaces*E*and*F*is an affine map*f*such that*f*(0_{E}) = 0_{F}. - A
*subspace*of a vector space*E*is an affine subspace of*E*containing 0_{E}(so by definition, each subspace is an affine subspace but not conversely) - Subspaces can be used in the role of directions, as any direction contains a unique subspace.

Indeed, affine subspaces

Now for this to form an equivalent presentation of the affine geometry of

- 0
_{E}∉*F*(gives surjectivity : the choice of 0_{E}is not structuring for*F*). - Then
*F*must be an hyperplane (to give injectivity).

A fundamental feature of affine geometry is that for any

The barycenter of points

Middles of segments are defined from barycenters.

A subset *F* of an affine space *E*
is a subspace of *E*, if and only if ∀*a,b*∈*F*,*
a*≠*b* ⇒ *a*∨*b* ⊂ *F*.

An equivalent condition for *F* ⊂ *E* to be an hyperplane is ∃*a*∈*E*\*F*, *E* = *F*∨*a*, which implies
∀*a*∈*E*\*F*, *E* = *F*∨*a*.

Generally ∀*F* ∈*S*(*E*), ∀*a*∈*E*\*F*,
Dim(*F*∨*a*) = Dim(*F*)+1.

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

It is a pair of sets (

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)

Let us see
how the previously mentioned pathological *D* is dismissed, not only when
viewed from a set theory without AC (but only DC), but by the duality structure itself which
can somehow express the idea of rejecting AC regardless the rest of universe.

Up: 5. Geometry

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