# continuous operator

## continuous operator

[kən¦tin·yə·wəs ′äp·ə‚rād·ər]
(mathematics)
A linear transformation of Banach spaces which is continuous with respect to their topologies.
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References in periodicals archive ?
In order words, [??] is continuous if and only if it is a component-wise continuous operator.
We say that the continuous operator F : [H.sup.n]([D.sub.V]) [right arrow] H([D.sub.V]) belongs to the class [U.sub.n,V] if, for every polynomial p = p(s),
This equality by (2.18) proves the existence of the continuous operator [(P - [lambda]I).sup.-1], which satisfies
Then, for any 0 < a < b < +[infinity], T : P [subset] ([bar.[Q.sub.b]]\[[OMEGA].sub.a]) [right arrow] P is a completely continuous operator.
This proves that [F.sub.i] is a continuous operator on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all i = 1,2.
iii) f : C(I, [R.sup.n]) [right arrow] C(I, [R.sup.n]) is a continuous operator for which there exist [[beta].sub.0] > 0, a nondecreasing function [beta] : [0, [infinity]) [right arrow] (0, [infinity]) and a bounded function [w.sub.1]: [0, [infinity]) [right arrow] [0, [infinity]) with [lim.sub.t[right arrow]0]+[w.sub.1](t) = 0 such that |(fx)(t)| [less than or equal to] [[beta].sub.0]|x|c [for all]x [member of] C(I, [R.sup.n]), t [member of] I and
They do not need initial or continuous operator training, operator intervention, or management of the process.
Assume [[OMEGA].sub.1], [[OMEGA].sub.2] are bounded open subsets of E with de [[OMEGA].sub.1], [[bar.[OMEGA]].sub.1] [subset] [[OMEGA].sub.2], and let T: K [intersection] ([[bar.[OMEGA]].sub.2]\[[OMEGA].sub.1]) [right arrow] K be a completely continuous operator such that one of the following holds:
Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the Lipschitz continuous operator with constant [delta] = r/r - r'.
Keywords: Time scale, delta and nabla derivatives and integrals, Green's function, completely continuous operator, eigenfunction expansion.

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