• 대한수학회지
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• 제54권1호
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• pp.319-357
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• 2017

There are presented certain results on extending continuous linear operators defined on spaces of E-valued continuous functions (defined on a compact Hausdorff space X) to linear operators defined on spaces of E-valued measurable functions in a way such that uniformly bounded sequences of functions that converge pointwise in the weak (or norm) topology of E are sent to sequences that converge in the weak, norm or weak* topology of the target space. As an application, a new description of uniform closures of convex subsets of C(X, E) is given. Also new and strong results on integral representations of continuous linear operators defined on C(X, E) are presented. A new classes of vector measures are introduced and various bounded convergence theorems for them are proved.

• 대한수학회보
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• 제59권5호
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• pp.1093-1103
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• 2022

In this paper, we establish an evaluation formula to calculate the Wiener integral of polynomials in terms of natural dual pairings on abstract Wiener spaces (H, B, 𝜈). To do this we first derive a translation theorem for the Wiener integral of functionals associated with operators in 𝓛(B), the Banach space of bounded linear operators from B to itself. We then apply the translation theorem to establish an integration by parts formula for the Wiener integral of functionals combined with operators in 𝓛(B). We finally apply this parts formula to evaluate the Wiener integral of certain polynomials in terms of natural dual pairings.

• 대한수학회보
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• 제46권2호
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• pp.263-279
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• 2009

On the setting of the unit ball of ${\mathbb{R}}^n$, we consider a Banach space of harmonic functions motivated by the atomic decomposition in the sense of Coifman and Rochberg [5]. First we identify its dual (resp. predual) space with certain harmonic function space of (resp. vanishing) logarithmic growth. Then we describe these spaces in terms of boundedness and compactness of certain Toeplitz operators.

• 대한수학회보
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• 제38권4호
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• pp.657-662
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• 2001

It is proved that if for each n, $1\leqp_n\leq2 \;and \;the(p_n, p’_n)$ Clarkson inequality holds in each Banach space X$_{n}$ then the (t, t’) Clarkson inequality holds in ($\sum^\infty_{n=1}\; X_n)_r, \;the \ell^r-sum \;of\; X_n’s,\; where\; 1\leqr<\infty,\; t=min{p, r, r’} \;and \;p = \;inf{p_n}.$ The (p, p’) Clarkson inequality is preserved by quotient maps and a new proof of a Takahashi-Kato theorem stating that the (p, p’) Clarkson inequality holds in a Banach space X if and only if it holds in its dual space $X_*$ is given.n.

• 대한수학회보
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• 제29권1호
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• pp.81-87
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• 1992

Wellknown invariance of domain theorems are Brower's invariance of domain theorem for continuous mappings defined on a finite dimensional space and Schauder-Leray's invariance of domain theorem for the class of mappings I+C defined on a infinite dimensional Banach space with I the identity and C compact. The two classical invariance of domain theorems were proved by applying the homotopy invariance of Brower's degree and Leray-Schauder's degree respectively. Degree theory for some class of mappings is a useful tool for mapping theorems. And mapping theorems (or surjectivity theorems of mappings) are closely related with invariance of domain theorems for mappings. In[4, 5], Browder and Petryshyn constructed a multi-valued degree theory for A-proper mappings. From this degree Petryshyn [9] obtained some invariance of domain theorems for locally A-proper mappings. Recently Browder [6] has developed a degree theory for demicontinuous mapings of type ( $S_{+}$) from a reflexive Banach space X to its dual $X^{*}$. By applying this degree we obtain some invariance of domain theorems for demicontinuous mappings of type ( $S_{+}$). ( $S_{+}$).

• 대한수학회지
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• 제45권5호
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• pp.1393-1404
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• 2008

Let K be a nonempty closed convex subset of a Banach space E. Suppose $\{T_{n}\}$ (n = 1,2,...) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that ${\cap}_{n=1}^{\infty}$ F$\(T_n){\neq}{\phi}$. For $x_0{\in}K$, define $x_{n+1}={\lambda}_{n+1}x_{n}+(1-{\lambda}_{n+1})T_{n+1}x_{n},n{\geq}0$. If ${\lambda}_n{\subset}[0,1]$ satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n=0$, we proved that $\{x_n\}$ weakly converges to some $z{\in}F\;as\;n{\rightarrow}{\infty}$ in the framework of reflexive Banach space E which satisfies the Opial's condition or has $Fr{\acute{e}}chet$ differentiable norm or its dual $E^*$ has the Kadec-Klee property. We also obtain that $\{x_n\}$ strongly converges to some $z{\in}F$ in Banach space E if K is a compact subset of E or there exists one map $T{\in}\{T_{n};n=1,2,...\}$ satisfy some compact conditions such as T is semi compact or satisfy Condition A or $lim_{n{\rightarrow}{\infty}}d(x_{n},F(T))=0$ and so on.

• 충청수학회지
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• 제3권1호
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• pp.121-124
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• 1990

In this paper, we introduce the notion of the compact range property and $weak^*$ Radon Nikodym property. We prove that the compact range property, weak Radon Nikodym property and $weak^*$ Radon Nikodym property in dual Banach space are all equivalent. Other related results and discussed.

• 대한수학회보
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• 제26권2호
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• pp.151-158
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• 1989

Let (H, B, .nu.) be an abstract Wiener space where H is a separable Hilbert space with the inner product <.,.> and the norm vertical bar . vertical bar=.root.<.,.>, which is densely and continuously imbedded into a separable Banach space B with the norm ∥.∥ , and .nu. is a probability measure on the Borel .sigma.-algebra B(B) of B which satisfies (Fig.) where $B^{*}$ is the topological dual of B and (.,.) is the natural dual pairing between B and $B^{*}$. We will regard $B^{*}$.contnd.H.contnd.B in the natural way. Thus we have =(y, x) for all y in $B^{*}$ and x in H. Let $R^{n}$ and C denote the n-dimensional Euclidean space and the complex numbers respectively.ctively.

• 대한수학회지
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• 제41권1호
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• pp.107-130
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• 2004

A $JB^{*}-triple$ is a Banach space A on which the group Aut(B) of biholomorphic automorphisms acts transitively on the open unit ball B of A. In this case, a triple product {$\cdots$} from $A\;\times\;A\;\times\;A\;to\;A$ can be defined in a canonical way. If A is also the dual of some Banach space $A_{*}$, then A is said to be a JBW triple. A projection R on A is said to be structural if the identity {Ra, b, Rc} = R{a, Rb, c, }holds. On $JBW^{*}-triples$, structural projections being algebraic objects by definition have also some interesting metric properties, and it is possible to give a full characterization of structural projections in terms of the norm of the predual $A_{*}$ of A. It is shown, that the class of structural projections on A coincides with the class of the adjoints of neutral GL-projections on $A_{*}$. Furthermore, the class of GL-projections on $A_{*}$ is naturally ordered and is completely ortho-additive with respect to L-orthogonality.

• 대한수학회보
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• 제33권2호
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• pp.277-287
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• 1996

The theory of integration of functions with values in a Banach space has long been a fruitful area of study. In the eight years from 1933 to 1940, seminal papers in this area were written by Bochner, Gelfand, Pettis, Birhoff and Phillips. Out of this flourish of activity, two integrals have proved to be of lasting: the Bochner integral of strongly measurable function. Through the forty years since 1940, the Bochner integral has a thriving prosperous history. But unfortunately nearly forty years had passed until 1976 without a significant improvement after B. J. Pettis's original paper in 1938 [cf. 11].