Vector spaces in duality
It is not necessary to assume the concept of a vector space for
simply and directly defining the notion of a pair of vector spaces
in duality, in the following way:
A pair of dual vector spaces, is a particular case of duality system.
It is a pair of sets (E,E'), where the elements of E
are called vectors, those of E' are called covectors
(or linear forms) together with an operation of scalar
product from E×E' to ℝ : for each u∈E and x∈E', we have u⋅x∈ℝ,
and that is subject to the following axioms:
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 ℝE', and a map from E
' to ℝE, are
injective; thus E plays the role of (is identifiable with)
a subset of ℝE ',
and E' plays the role of a subset of ℝE
'as identified to its image in ℝE.
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'.
- ∀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)
A subspace is a subalgebra
(i.e. stable) for addition and multiplication by a number.
Note that (as for any duality
system) from any pair (E,E') with an operation from E×E' to ℝ
we can produce a pair of spaces in duality by replacing E by the
vector space generated by its image in ℝE', and the same
for E'. This can be done in any order without affecting the result
and indeed gives a pair of dual vector spaces by doing this
replacement just once on each side, because this replacement for E
does not affect the linear relations in E' (relations in E' that are
true in ℝE remain true when replacing E by the vector
space generated by its image in ℝE'), thanks to the
algebraic relations between addition and multiplication
(commutativity, associativity, distributivity).
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)
Inner product spaces
While the value of the dimension n sufficed to classify
affine geometries, it does not suffice with inner products. Inner
products are classified by an oriented pair of natural numbers (p,q)
with p+q=n, called the signature,
which roughly means that "scalar squares are positive in p
dimensions and negative in q dimensions". The geometry is
Euclidean if scalar squares of all vectors have the same sign
(usually taken as positive, that is q=0, but the opposite
sign convention of negative scalar squares, that is p=0,
essentially gives the same geometry).
Notice that the set of "rotations" (transformations which preserve
the inner product) has dimension n(n-1)/2 for any signature.
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