# strong topology

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## 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 ?
where [S.sup.*](Z) is the dual of S(Z), which is endowed with the strong topology, which cannot be induced by any norm [5].
Let [S.sup.*](Z) be the dual of S(Z) and endow it with the strong topology. Then,
and moreover the inductive limit topology over [S.sup.*](Z) given by space sequence [{[S.sup.*.sub.p](Z)}.sub.p[greater than or equal to]0] coincides with the strong topology.
The remainder of the book proceeds from this theme, discussing equicontinuity, the strong topology, operators, completeness, inductive limits, compactness, and barrelled spaces.
There is a natural topology, called the strong topology, on each PN space.
The strong topology on a PN space (V, v, [tau], [[tau].sup.*]) is determined by the system of neighborhoods
In this case, we will call this mapping an LS mapping, means the linearly homeomorphic with respect to the strong topology, in order to distinguish the notion of the linearly homeomorphic in topological vector space in which the linear structure and the topological structure are compatible with each other.
Then the mapping f [??] P([[DELTA].sub.[alpha],[beta]])f is linear and continuous from [H'.sub.[alpha],[beta],k] into itself when in [H'.sub.[alpha],[beta],k] we consider either the weak or the strong topology.
Theorem 4.2: (Continuity of #-convolution): Let k[member of]Z,k<0, Assume that [([f.sub.n]).sub.n[member of]N], is a sequence in [h.sub.'[alpha],[beta],k] that converges to f [member of] [h.sub.'[alpha],[beta],k] in the weak topology (respectively in the strong topology) of [h.sub.'[alpha],[beta],k].
Its dual space [F'.sub.[theta]](N'), equipped with the strong topology, is called the space of distributions on N'.
*[PHI] (n times) and the convergence is in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (N') with respect to the strong topology.
Let E be the topological dual of [C.sub.c](X) and denote by B the family of all bounded sets of [C.sub.c] (X) and by [beta](E, C(X)) the strong topology on E.
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