# strong topology

(redirected from Strongly continuous)

## strong topology

[′strȯŋ tə′päl·ə·jē]
(mathematics)
The topology on a normed space obtained from the given norm; the basic open neighborhoods of a vector x are sets consisting of all those vectors y where the norm of x-y is less than some number.
References in periodicals archive ?
A strongly continuous vector function y : I [right arrow] X is called a weak solution of the evolution equation
(ii) Let [([R.sub.1](t)).sub.t[member of][0,[tau])] [subset or equal to] L(Y,X) be strongly continuous. Then it is said that A is a subgenerator of a (local, if [tau] < [infinity]) mild (a, k)-regularized [C.sub.1] -existence family [([R.sub.1](t)).sub.t[member of][0,[tau])] iff (25) holds.
Lately, there has been an increasing interest in the study of the exponential dichotomy of the exponentially bounded, strongly continuous cocycles over continuous flows on a locally compact metric space [THETA] and acting on a Banach space X (see for instance ).
A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators [(C(t)).sub.t[member of]R] on X.
where A generates a strongly continuous, not necessarily compact, semigroup [(T(t)).sub.t[greater than or equal to]0] in the Banach space x.
The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow.
 if A is a closed linear operator with domain D(A) defined on a Banach space E and [alpha] > 0, the we say that A is the generator of an [alpha]-resolvent family if there exists [omega] [greater than or equal to] 0 and a strongly continuous function [S.sub.[alpha]] : [R.sub.+] [right arrow]L(E) such that {[[lambda].sub.[alpha]] : Re([lambda]) > [omega]} [subset] [rho](A)) ([rho](A) being the resolvent set of A) and
Motivated by the works of Mustari and Taylor [15,17], we have recently begun to study the mean ergodic theorem under the framework of RN modules  to obtain the mean ergodic theorem in the sense of convergence in probability, where we proved the mean ergodic theorem for a strongly continuous semigroup of random unitary operators defined on complete random inner product modules (briefly, complete RIP modules).
The derivative [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and it is strongly continuous on 0 [less than or equal to] [theta] [less than or equal to] t [less than or equal to] T.
Among his topics are strongly continuous semigroups, compatible and regular systems, stabilization and detection, and time discrete systems.
So, X is weakly compact and, at the same time, [PHI] is weakly continuous, being affine and strongly continuous.
Levine (1960) introduced strongly continuous functions in the widely read Classroom Notes section of the American Mathematical Monthly.
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