• Title, Summary, Keyword: Almost linear space

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COMPLETENESS OF A NORMED ALMOST LINEAR SPACE B(X, (Y,C))

  • Lee, Sang Han;Im, Sung Mo
    • Journal of the Chungcheong Mathematical Society
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    • v.13 no.1
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    • pp.79-85
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    • 2000
  • In this paper, we have an affirmative solution of G. Godini's open question ([3]): If a normed almost linear space Y is complete, is the normed almost linear space B(X, (Y,C)) complete?

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A METRIC ON NORMED ALMOST LINEAR SPACES

  • Lee, Sang-Han;Jun, Kil-Woung
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.2
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    • pp.379-388
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    • 1999
  • In this paper, we introduce a semi-metric on a normed almost linear space X via functional. And we prove that a normed almost linear space X is complete if and only if $V_X$ and $W_X$ are complete when X splits as X=$W_X$ + $V_X$. Also, we prove that the dual space $X^\ast$ of a normed almost linear space X is complete.

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BASIS FOR ALMOST LINEAR SPACES

  • Lee, Sang-Han
    • The Pure and Applied Mathematics
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    • v.2 no.1
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    • pp.43-51
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    • 1995
  • In this paper, we introduce the almost linear spaces, a generalization of linear spaces. We prove that if the almost linear space X has a finite basis then, as in the case of a linear space, the cardinality of bases for the almost linear space X is unique. In the case X = Wx + Vx, we prove that B'= {$\chi$'$_1,...,x'_n} is a basis for the algebraic dual X$^#$ of X if B = {$\chi$'$_1,...,x'_n} is a basis for the almost linear space X. And we have an example X($\neq$Wx + Vx) which has no such a basis.

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A STRONG LAW OF LARGE NUMBERS FOR AANA RANDOM VARIABLES IN A HILBERT SPACE AND ITS APPLICATION

  • Ko, Mi-Hwa
    • Honam Mathematical Journal
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    • v.32 no.1
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    • pp.91-99
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    • 2010
  • In this paper we introduce the concept of asymptotically almost negatively associated random variables in a Hilbert space and obtain the strong law of large numbers for a strictly stationary asymptotically almost negatively associated sequence of H-valued random variables with zero means and finite second moments. As an application we prove a strong law of large numbers for a linear process generated by asymptotically almost negatively random variables in a Hilbert space with this result.

Uniqueness of Bases for almost linear spaces

  • Im, Sung-Mo;Lee, Sang-Han
    • Bulletin of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.127-133
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    • 1997
  • O. Mayer[9] introduced an almost linear space (als), a generalization of a linear space. The notion of a basis for an als was introduced by G. Godini[3]. Later, man properties of an als established by a number of authors. In this paper, we prove that the cardinality of bases for an als is unique. All spaces involved in this paper are over the real field $R$. Let us denote by $R_+$ the set ${\lambda \in R : \lambda \geq 0}$. We recall some definitions used in this paper.

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REFLEXIVITY OF NORMED ALMOST LINEAR SPACES

  • Lee, Sang-Han
    • Communications of the Korean Mathematical Society
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    • v.10 no.4
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    • pp.855-866
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    • 1995
  • We prove that if a nals X is reflexive, then $X = W_X + V_X$. We prove also that if an als X has a finite basis, then $X = W_X + V_X$ if and only if X is reflexive.

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A metric induced by a norm on normed almost linear spaces

  • Im, Sung-Mo;Lee, Sang-Han
    • Bulletin of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.115-125
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    • 1997
  • In [3,4,5], G. Godini introduced a normed almost linear space(nals), generalizing the concept of a normed linear space. In contrast with the case of a normed linear space, tha norm of a nals $(X, $\mid$$\mid$$\mid$ \cdot $\mid$$\mid$$\mid$)$ does not generate a metric on X $(for x \in X \backslash V_X we have $\mid$$\mid$$\mid$ x - x $\mid$$\mid$$\mid$ \neq 0)$.

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A CHARACTERIZATION OF REFLEXIVITY OF NORMED ALMOST LINEAR SPACES

  • Im, Sung-Mo;Lee, Sang-Han
    • Communications of the Korean Mathematical Society
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    • v.12 no.2
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    • pp.211-219
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    • 1997
  • In [6] we proved that if a nals X is reflexive, then $X = W_X + V_X$ . In this paper we show that, for a split nals $X = W_X + V_X$, X is reflecxive if and only if $V_X$ and $W_X$ are reflcxive.

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SOME CONVERGENCE THEOREM FOR AND RANDOM VARIABLES IN A HILBERT SPACE WITH APPLICATION

  • Han, Kwang-Hee
    • Honam Mathematical Journal
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    • v.36 no.3
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    • pp.679-688
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    • 2014
  • The notion of asymptotically negative dependence for collection of random variables is generalized to a Hilbert space and the almost sure convergence for these H-valued random variables is obtained. The result is also applied to a linear process generated by H-valued asymptotically negatively dependent random variables.