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Let $X$, $Y$, $Z$ normed spaces If X$\cong Y\oplus Z$ why is $X^*\cong Y^*\oplus Z^*$? where $X^*$ is the dual of $X$. For example ${\ell^\infty}^*\cong\ell^1\oplus\mathrm{Null}\;C_0$ so if we take the double dual we find the ${\ell^\infty}^{**}\cong{\ell^1}^*\oplus (\mathrm{Null}\; C_0)^*$. I am not sure I understand why these equalities should hold?

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    Do you understand the corresponding fact for vector spaces?2012-10-20

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