RUANG MATRIX LINEAR TRANSLASI INVARIAN PADA RUANG FUNGSI INTEGRAL HENSTOCK-DUNFORD PADA [a,b]

Abstract

In this paper we study Henstock-Dunford integral on [a,b]. We discuss some properties of the integrable. We will construct norm and matrix on Dunford-Henstock integrable function space, $HD[a,b]$. We obtain that $HD[a,b]$ is linear space. A function $\left\| \,.\, \right\|:HD[a,b]\to R$ defined by $\left\| f \right\|=\underset{\begin{smallmatrix} {{x}^{*}}\in {{X}^{*}} \\ \left\| {{x}^{*}} \right\| \le 1 \end{smallmatrix}}{\mathop{\sup}}\, \\ left( \underset{A\subset[a,b]}{\mathop{\sup}}\,\,\left| \left( H \right) \int \limits_{A}{{{x}^{*}}f} \right| \right)$ for every $f \in HD[a,b]$ is norm on linear space $HD[a,b]$. A function $d:HD[a,b]\times HD[a,b]\to R$ defined by $d\left( f,g \right)=\left\| f-g \right\|$ for every $f,g\in HD[a,b]$ is a matrix on linear space $HD[a,b]$. Further more, linear space $HD[a,b]$ is linear matrix translation invarian space.