• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.239-244
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• 1993

The purpose of this paper is to determine conditions under which equicontinuity of the family of iterates {f$^{n}$ } of a continuous function that maps the circle S$^{1}$ into itself does occur. We shall see that equicontinuity of the family of iterates {f$^{n}$ } occurs only under special cases. Actually, we will show that this happens only for rotations when degree of the function is 1, and for involutions when degree of the function is -1.

• Journal of the Korean Statistical Society
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• v.24 no.1
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• pp.225-231
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• 1995

An eventual uniform equicontinuity condition is investigated in the context of the uniform central limit theorem (UCLT) for continuous martingales. We assume the usual intergrability condition on metric entropy. We establish an exponential inequality for a martingales. Then we use the chaining lemma of Pollard (1984) to prove an eventual uniform equicontinuity which is a sufficient condition of UCLT. We apply the result to approximate a stochastic integral with respect to a martingale to that of a Brownian motion.

• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.643-656
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• 2009

The family of demi-linear mappings between topological vector spaces is a meaningful extension of the family of linear operators. We establish equicontinuity results for demi-linear mappings and develop the usual theory of distributions and the usual duality theory.

• Communications of the Korean Mathematical Society
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• v.2 no.1
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• pp.123-128
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• 1987

• Kyungpook Mathematical Journal
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• v.28 no.1
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• pp.23-34
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• 1988

A version of Ascoli's Theorem (equating compact and equicontinuous sets) is presented in the context of convergence spaces. This theorem and another, (involving equicontinuity) are applied to characterize compact subsets of quasi-multipliers of a $C^*$-algebra B, and to characterize the compact subsets of the state space of B. The classical Ascoli Theorem states that, for pointwise pre-compact families F of continuous functions from a locally compact space Y to a complete Hausdorff uniform space Z, equicontinuity of F is equivalent to relative compactness in the compact-open topology([4] 7.17). Though this is one of the most important theorems of modern analysis, there are some applications of the ideas inherent in this theorem which arc not readily accessible by direct appeal to the theorem. When one passes to so-called "non-commutative analysis", analysis of non-commutative $C^*$-algebras, the analogue of Y may not be relatively compact, while the conclusion of Ascoli's Theorem still holds. Consequently it seems plausible to establish a more general Ascoli Theorem which will directly apply to these examples.

• Korean Journal of Mathematics
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• v.18 no.3
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• pp.253-259
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• 2010

In this paper we establish approximation of contraction C-semigroups on the extrapolation space $X^C$, by showing the equicontinuity of contraction C-semigroups on $X^C$.

• Korean Journal of Mathematics
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• v.22 no.2
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• pp.349-354
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• 2014

In this paper, we establish convergence of contraction $C_0$ semigroups in the weak topology on a general Banach space. We remove the restriction on a Banach space X and weaken the condition on resolvents of generators in the previous results [4, 5].

• Communications of the Korean Mathematical Society
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• v.14 no.3
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• pp.581-595
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• 1999

In the present paper we provide a short of the uni-form CLT for the function-indexed martingale difference process under the uniformly integrable entropy by establishing a maximal inequality.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.427-446
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• 1995

Let S be a set and B be a $\sigma$-field on S. We consider $(\Omega = S^Z, T = B^z, P)$ as the basic probability space. We denote by T the left shift on $\Omega$. We assume that P is invariant under T, i.e., $PT^{-1} = P$, and that T is ergodic. We denote by $X = \cdots, X_-1, X_0, X_1, \cdots$ the coordinate maps on $\Omega$. From our assumptions it follows that ${X_i}_{i \in Z}$ is a stationary and ergodic process.

• Kyungpook Mathematical Journal
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• v.28 no.1
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• pp.79-82
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• 1988

This paper investigates the relationship between compactness and pseudo-compactness in subsets of C(X) where X is locally compact and first countable. Two primary theorems are proven. First, equicontinuity at a point is proven to be equivalent to the existence of a certain open cover of a pseudo-compact subset of C(X). The second theorem proves the equivalence of compactness and pseudo-compctness for closed subsets F of C(X).