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Let $X$ be a compact ball in $\mathbb{R}^n$. Let $C_0^k(X)$ be the space of a $k$ times continuously differentiable complex, valued smooth functions, which vansih outside of $X$.

The norm on $C_0^k(X)$ is given by $$ ||f|| = \sum_{| \alpha | \leq k} || \partial^\alpha f ||_\infty $$

How does the Banach space dual space of $C_0^k(X)$ look?

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    $\begingroup$ The purely algebraic dual? $\endgroup$
    – Rasmus
    Commented Nov 13, 2011 at 18:16
  • $\begingroup$ ;) No, the topological one. $\endgroup$
    – Marc Palm
    Commented Nov 13, 2011 at 19:02

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If I remember correctly, it is isomorphic to the direct sum of the dual of $C_0(X)$ and a $k$-dimensional vector space (the precise answer is in "Linear Operators", Dunford/Schwartz).

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  • $\begingroup$ I will check the reference. Something similar was also my guess. Thanks $\endgroup$
    – Marc Palm
    Commented Nov 14, 2011 at 10:20

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