Download Optimal Control of Differential and Functional Equations by J. Warga PDF

By J. Warga

ISBN-10: 0127351507

ISBN-13: 9780127351506

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Example text

Then lim, Ιόχ = f(x) for all xeX. I PROOF Let x G #■ and e > 0. Then there are y0 G N and j G Χλ such that l/(*) - / ω ΐ < €/3, | j> - x | < e/3, and | / ( j ) - l, y \ < e/3 for ally > y 0 . Thus |/(x) - Ice I < |/(*) -f(y)\ + \fiy) - hy | + | /,Ο - x)\ < e for all j^j0. 11 Theorem (Bishop) Let X be a separable normed vector space and £/ Δ {/G#** | | /1 Δ s u p i ^ i | l(x)\ < 1}. Then there exists a norm I · \w (weak norm) on X* such that the relativizations to U of the corresponding metric topology (weak metric topology) and of the weak star topology of X* coincide, and lim, | / — /,,\w = 0 for /, /, G U if and only if lim,· IjX = Ix for each x e l I PROOF Let Χ^ Δ {jq , JC2 v··} be a dense subset of #" and n I Γ L Δ lim X 2"' | Tx5 |/(1 + | x,· |) n j=i for all Γ G X*.

And their diagonal subsequence (/,)3e

Ste/? 3 We now prove that for every A e Σ there exists A' C A with μ,(Λί') = \μ(Α). If M-4) = 0, then A' A A yields μ(Α') = \μ(Α). We therefore assume that μ{Α) > 0. , C%m) of C with 0 < MQ m ) < 1/ra.

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