SEMINORM PADA RUANG FUNGSI TERINTEGRAL DUNFORD

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