• Title, Summary, Keyword: Normed almost linear space

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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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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 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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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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LINEAR MAPPINGS, QUADRATIC MAPPINGS AND CUBIC MAPPINGS IN NORMED SPACES

  • Park, Chun-Gil;Wee, Hee-Jung
    • The Pure and Applied Mathematics
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    • v.10 no.3
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    • pp.185-192
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    • 2003
  • It is shown that every almost linear mapping $h{\;}:{\;}X{\;}{\rightarrow}{\;}Y$ of a complex normed space X to a complex normed space Y is a linen. mapping when h(rx) = rh(x) (r > 0,$r\;{\neq}\;1$$x{\;}{\in}{\;}X$, that every almost quadratic mapping $h{\;}:{\;}X{\;}{\rightarrow}{\;}Y$ of a complex normed space X to a complex normed space Y is a quadratic mapping when $h(rx){\;}={\;}r^2h(x){\;}(r{\;}>{\;}0,r\;{\neq}\;1)$ holds for all $x{\;}{\in}{\;}X$, and that every almost cubic mapping $h{\;}:{\;}X{\;}{\rightarrow}{\;}Y$ of a complex normed space X to a complex normed space Y is a cubic mapping when $h(rx){\;}={\;}r^3h(x){\;}(r{\;}>{\;}0,r\;{\neq}\;1)$ holds for all $x{\;}{\in}{\;}X$.

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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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