# bounded sequence

## bounded sequence

[′bau̇nd·əd ′sē·kwəns]
(mathematics)
A sequence whose members form a bounded set.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Since the left hand side is the member of a bounded sequence, while the right hand side approaches +[infinity], we have a contradiction.
(ii) In addition, if {[x.sub.n]} c E is a bounded sequence such that [lim.sub.n[right arrow][infinity]] dist([x.sub.n], T([x.sub.n])) = 0, where dist(a, B) is the distance from a point a e X to the set B e C(X), then for all Banach limits [mu],
Let ([x.sub.n]) [subset] X be a bounded sequence. It is well known and easy to see that ([x.sub.n]) defines an operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] through the equality
where [f.sub.n] (t,x, [lambda]) = f (*, *,[lambda]) * [n.sup.2][omega](nt, nx) is a regularization of the flux f via the standard non-negative mollifier [omega] [member of] [C.sup.[infinity].sub.c][((-1,1).sup.2]) with total mass one, and ([u.sup.0.sub.n]) is a bounded sequence of functions converging strongly in [L.sup.1.sub.loc] (R) toward [u.sub.0] By multiplying (1.4) by sgn([u.sub.n]t,x) - [lambda]) we get after standard manipulations (see also formula (2.8) of the next section):
Assume here and after that x = ([x.sub.k]) be a sequence such that [x.sub.k] [not equal to] 0 for all k [member of] N and ([p.sub.k]) be the bounded sequence of strictly positive real numbers with [supp.sub.k] = H and M = max{1, H}.
For any bounded sequence g(n) defined on Z, define [g.sup.u] = [sup.sub.n[member of]Z] g(n), [g.sup.l] = [inf.sub.n[member of]Z] g(n).
In other words [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a bounded sequence.
where {[a.sub.n]}, {[b.sub.n]}, {[c.sub.n]} are sequences in [0,1] such that [a.sub.n] + [b.sub.n] + [c.sub.n] = 1 and {[u.sub.n]} is a bounded sequence in D(T).
Let [lambda] [member of] [OMEGA] and let x = ([x.sub.k]) be a bounded sequence in X.
In other words (M([[|[x.sub.k]|.sup.1/k]/[rho]])) is a bounded sequences. [[conjunction].sub.M] is called the Orlicz space of bounded sequence.
Then for every bounded sequence {[{[[omega].sub.j]}.sub.j[member of]J], for all [lambda] [member of] [0,1] and all f [member of] H, we have
Let [bar.x] = [([x.sub.n]).sub.n] [subset or equal to] X be an f--statistically bounded sequence, then the set [[GAMMA].sup.f.sub.[bar.x]] is bounded.

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