Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
The weak-* topology on L ∞ sets
A.3 Uniform Continuity of the Resolvant
According to what is said in Section A.2, we can embed L ∞ with either the weak or the
weak-∗ topology. We choose the weak-∗ topology, the coarser one 8 . The following theorem,
stated with the same notations as before, gives us an important property of this topology:
Theorem 73 ([33], III-25, Pg 48)
Let E be a separable Banach space, then BE ∗ = {f ∈ E∗ ; �f� ≤1} is metrizable for the
weak-∗ topology 9 .
Conversely, if BE ∗ = {f ∈ E∗ ; �f� ≤1} is metrizable for the weak-∗ topology then E is
separable.
Now, recall that (L 1 ) ∗ = L ∞ and L ∞ ⊂ (L 1 ) ∗ [33, 107] and consider the three theorems
below.
Theorem 74 ([33])
I - IV-7, Pg 57: L p is a normed vector space for 1 ≤ p ≤∞.
II - (Fischer-Riesz) - IV-8, Pg 57: L p is a Banach space for 1 ≤ p ≤∞.
III - IV-13, Pg 62: L p is separable 10 for 1 ≤ p