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A Characterization of the Weak*-Integral
RHIE GIL SEOB,PARK HI KYO 충청수학회 1989 충청수학회지 Vol.2 No.1
The main goal of the present paper is to characterize the weak*-integral, which is a weak* analogy of Geitz[4]
Some properties of the convergence of sequences of fuzzy points in a fuzzy normed linear space
Gil Seob Rhie,Young Uk Do 한국지능시스템학회 2007 한국지능시스템학회논문지 Vol.17 No.1
With a new ordinary norm as an analogy of Krishna and Sarma[5] and Bag and Samanta[1], we will characterize the notions of the convergence of the sequences of fuzzy points, the fuzzy α-Cauchy sequence and fuzzy completeness.
On the fuzzy convergence of sequences in a fuzzy normed linear space
Gil Seob Rhie,In Ah Hwang 한국지능시스템학회 2008 한국지능시스템학회논문지 Vol.18 No.2
In this paper, we introduce the notions of a fuzzy convergence of sequences, fuzzy Cauchy sequence and the related fuzzy completeness on a fuzzy normed linear space. And we investigate some properties relative to fuzzy normed linear spaces. In particular, we prove an equivalent conditions that a fuzzy norm defined on a ordinary normed linear space is fuzzy complete.
A Characterization of the Weak<SUP>*</SUP>-Integral
Rhie, Gil-Seob,Park, Hi-Kyo 충청수학회 1989 충청수학회지 Vol.2 No.1
The main goal of the present paper is to characterize the $weak^*$-integral, which is a $weak^*$ analogy of Geitz[4].
Some properties of equivalent fuzzy norms
Rhie, Gil-Seob,Hwang, In-Ah Korean Institute of Intelligent Systems 2005 INTERNATIONAL JOURNAL of FUZZY LOGIC and INTELLIGE Vol.5 No.2
In the present paper, we observe a relation between fuzzy norms and induced crisp norms on a linear space. We first prove that if $\rho_1,\;\rho_2$ are equivalent fuzzy norms on a linear space, then for every $\varepsilon\in(0.1)$, the induced crisp norms $P_\varepsilon^1,\;and\;P_\varepsilon^2$, respectively are equivalent. Since the converse does not hold, we prove it under some strict conditions. And consider the following theorem proved in [8]: Let $\rho$ be a lower semicontinuous fuzzy norm on a normed linear space X, and have the bounded support. Then $\rho$ is equivalent to the fuzzy norm $\chi_B$ where B is the closed unit ball of X. The lower semi-continuity of $\rho$ is an essential condition which guarantees the continuity of $P_\varepsilon$, where 0 < e < 1. As the last result, we prove that : if $\rho$ is a fuzzy norm on a finite dimensional vector space, then $\rho$ is equivalent to $\chi_B$ if and only if the support of $\rho$ is bounded.
Some properties of equivalent fuzzy norms
Gil Seob Rhie,In Ah Hwang 한국지능시스템학회 2005 INTERNATIONAL JOURNAL of FUZZY LOGIC and INTELLIGE Vol.5 No.2
In the present paper, we observe a relation between fuzzy norms and induced crisp norms on a linear space. We first prove that if ρ₁, ρ₂ are equivalent fuzzy norms on a linear space, then for every ε∈ ( 0, 1 ) , the induced crisp norms P¹ε and P²ε respectively are equivalent. Since the converse does not hold, we prove it under some strict conditions. And consider the following theorem proved in [8]: Let ρ be a lower semicontinuous fuzzy norm on a normed linear space X , and have the bounded support. Then ρ is equivalent to the fuzzy norm χ B where B is the closed unit ball of X. The lower semi-continuity of ρ is an essential condition which guarantees the continuity of Pε where 0 < ε < 1 . As the last result, we prove that : if ρ is a fuzzy norm on a finite dimensional vector space, then ρ is equivalent to χB if and only if the support of ρ is bounded.
On the fuzzy convergence of sequences in a fuzzy normed linear space
Rhie, Gil-Seob,Hwang, In-Ah Korean Institute of Intelligent Systems 2008 한국지능시스템학회논문지 Vol.18 No.2
In this paper, we introduce the notions of a fuzzy convergence of sequences, fuzzy Cauchy sequence and the related fuzzy completeness on a fuzzy normed linear space. And we investigate some properties relative to fuzzy normed linear spaces. In particular, we prove an equivalent conditions that a fuzzy norm defined on a ordinary normed linear space is fuzzy complete.