SEMINORM PADA RUANG FUNGSI TERINTEGRAL DUNFORD

Solikhin Solikhin, Y.D. Sumanto, Abdul Aziz


DOI: https://doi.org/10.14710/jfma.v2i1.30

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 $d\left( f,g \right)=\left\| f-g \right\|=\underset{\begin{smallmatrix}  {{x}^{*}}\in {{X}^{*}} \\  \left\| {{x}^{*}} \right\|\le 1 \end{smallmatrix}}{\mathop{\sup }}\,\ \left\{ \underset{E\subset [a,b]}{\mathop{\sup }}\,\,\left| \left( L \right)\int\limits_{E}{{{x}^{*}}\left( f-g \right)} \right| \right\}$.

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References


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