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Adaptative high-gain extended Kalman filter and applications

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tel-00559107, version 1 - 24 Jan 2011<br />

The weak-* topology on L ∞ sets<br />

A.3 Uniform Continuity of the Resolvant<br />

According to what is said in Section A.2, we can embed L ∞ with either the weak or the<br />

weak-∗ topology. We choose the weak-∗ topology, the coarser one 8 . The following theorem,<br />

stated with the same notations as before, gives us an important property of this topology:<br />

Theorem 73 ([33], III-25, Pg 48)<br />

Let E be a separable Banach space, then BE ∗ = {f ∈ E∗ ; �f� ≤1} is metrizable for the<br />

weak-∗ topology 9 .<br />

Conversely, if BE ∗ = {f ∈ E∗ ; �f� ≤1} is metrizable for the weak-∗ topology then E is<br />

separable.<br />

Now, recall that (L 1 ) ∗ = L ∞ <strong>and</strong> L ∞ ⊂ (L 1 ) ∗ [33, 107] <strong>and</strong> consider the three theorems<br />

below.<br />

Theorem 74 ([33])<br />

I - IV-7, Pg 57: L p is a normed vector space for 1 ≤ p ≤∞.<br />

II - (Fischer-Riesz) - IV-8, Pg 57: L p is a Banach space for 1 ≤ p ≤∞.<br />

III - IV-13, Pg 62: L p is separable 10 for 1 ≤ p

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