By Zabczyk J.
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In particular, (18) follows from (13). As u is the uniform Iimit of the continuous functions Vn, it is also continuous. v, in the Iimit we have u(0)=11v, i. e. 2). ' We write tc D if t= (pjq) T with p, q E N.
Notice that the "obvious" choice s--+ proj(x, C(s)) rnay have unbounded variation. ( II u( t+c) 11 2 -II u(t) 11 2 ) = ~[u(t+c) + u( t)]. [u(t+c)-u(t)]. - = x-(x-<1>) and choose continuous at t+E, so that du([ t, t+c[) f in such a way that u is also = ·u( t+E )-u( t) = du([ t, t+E]). x. du([ t, t+c]) ;::: ~[ u( t+c) + u( t)]. du([ t,t+c]) - +j whence (x - ! [u( t+E) + u( t)] ) . du, I du I ([ t, t + f]) . I du I ([t, t+c]) (with the convention 0/0 = 0) and let c--+0, picking only continuity points t+c.
4 on the extraction of convergent subsequences, in the sense of filled-in graphs, does not apply here, tlms forcing some Ievel of complexity on the proof. 2. Preliminary estimates We aim at obtaining an upper bound of the Hilbert norm of that effect some approximations of uA . duA dt (t) using to Let v : I-; IR be the variation function of C (or more generally a "super-variation", i. 1 )) . 2 Prelirninary estimates 31 right-continuous on I and continuous at T: v-( T) = v( T). Hence, for every € > 0, it is possible to find a partition Pf of the interval I: io=O< t 1 < ...