Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
A.2 Weak-∗ Topology
The problem “find the coarser 4 topology that makes the applications ϕi continuous” is
transformed into: “find the coarser family of subsets of X that contains (Oλ)λ∈Λ and that is
stable under any finite intersection and any infinite union of its elements” 5 .
Consider first the family of all the subsets of X obtained as the intersection of a finite
number of elements of the form Oλ, λ ∈ Λ. It is a family of subsets of X stable under finite
intersections.
Secondly, consider all the infinite unions of elements of this latter family 6 . We finally end
up with the family
⎧
⎨
˜T =
⎩
�
infinite
⎡
⎣ �
finite
Oλ
⎤ ⎫
⎬
⎦ , λ ∈ Λ
⎭ .
˜T is a topology: stability under infinite union is obvious, and the proof of the stability under
finite intersection is left as an exercise of sets theory.
Consider now any topology ˆ T of X that makes all the applications ϕi : X → Yi, i ∈ I
continuous. Then all the subsets of the form ϕ −1
i (ωi), ωi being an open subset of Yi, i ∈ I, are
contained in ˆ T. And since ˆ T is a topology, it contains all the elements of ˜ T. Thus any topology
that makes all (ϕi)i∈I continuous contains ˜ T: it is the coarser topology we are searching for.
Definition of the Weak Topology
For a fixed f ∈ E ∗ , we define the application ϕf : E → R by ϕf (x) =< f, x >. When f
describes the set E ∗ , we have a family of applications (ϕf )f∈E ∗.
Definition 70
The weak topology is the coarser topology that renders all the applications (ϕf )f∈E ∗
continuous.
A weak topology is built on the dual space E ∗ by considering the applications (ϕξ)ξ∈E ∗∗.
Definition of the Weak-∗ Topology
We define a canonical injection from E to E ∗∗ :
− let x ∈ E be fixed
− E∗ → R
f ↦→ < f, x > is a continuous linear form on E∗ , that is to say an element of E ∗∗ ,
denoted Jx.
− we have:
< Jx, f > (E ∗∗ ,E ∗ )=< f, x > (E ∗ ,E), ∀x ∈ E, ∀f ∈ E ∗ .
The application x ↦→ Jx is a linear injection.
4
The coarser topology is the one that contains “the least” number of open sets.
5
Obviously ∅ and X are contained in (Oλ)λ∈Λ. Therefore the stability under infinite unions, and the
stability under finite intersections define a topology.
6
If we perform the infinite union first and then the finite intersection, we obtain a family that is not stable
under infinite union anymore.
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