SEMINORM PADA RUANG FUNGSI TERINTEGRAL DUNFORD

Solikhin Solikhin, Y. Sumanto, Abdul Aziz
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Abstract

This article discussed the seminorm on Dunford integrable functional space. We show that the set of all Dunford integrable functions is linear space. The results were shown that $\left( D[a,b],\ \left\| \ \cdot \  \right\| \right)$ is a seminorm space with function defined by $\left\| f \right\|=\underset{\begin{smallmatrix}  {{x}^{*}}\in {{X}^{*}} \\  \left\| {{x}^{*}} \right\|\le 1 \end{smallmatrix}}{\mathop{\sup }}\,\ \left\{ \underset{E\subset [a,b]}{\mathop{\sup }}\,\,\left| \left( L \right)\int\limits_{E}{{{x}^{*}}f} \right| \right\}$. Furthermore, $\left( D[a,b],\ d \right)$ is a pseudomatrix space with function defined by $d\left( f,g \right)=\left\| f-g \right\|=\underset{\begin{smallmatrix}  {{x}^{*}}\in {{X}^{*}} \\  \left\| {{x}^{*}} \right\|\le 1 \end{smallmatrix}}{\mathop{\sup }}\,\ \left\{ \underset{E\subset [a,b]}{\mathop{\sup }}\,\,\left| \left( L \right)\int\limits_{E}{{{x}^{*}}\left( f-g \right)} \right| \right\}$.
本文讨论了关于Dunford可积泛函空间的研讨会。我们证明了所有邓福德可积函数的集合是线性空间。结果表明:$\left( D[a,b],\ \left\| \ \cdot \  \right\| \right)$是一个半精空间,其函数定义为$\left\| f \right\|=\underset{\begin{smallmatrix}  {{x}^{*}}\in {{X}^{*}} \\  \left\| {{x}^{*}} \right\|\le 1 \end{smallmatrix}}{\mathop{\sup }}\,\ \left\{ \underset{E\subset [a,b]}{\mathop{\sup }}\,\,\left| \left( L \right)\int\limits_{E}{{{x}^{*}}f} \right| \right\}$。更进一步,$\left( D[a,b],\ d \right)$是一个伪矩阵空间,其函数定义为$d\left( f,g \right)=\left\| f-g \right\|=\underset{\begin{smallmatrix}  {{x}^{*}}\in {{X}^{*}} \\  \left\| {{x}^{*}} \right\|\le 1 \end{smallmatrix}}{\mathop{\sup }}\,\ \left\{ \underset{E\subset [a,b]}{\mathop{\sup }}\,\,\left| \left( L \right)\int\limits_{E}{{{x}^{*}}\left( f-g \right)} \right| \right\}$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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