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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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The purely algebraic dual? –  Rasmus Nov 13 '11 at 18:16
    
;) No, the topological one. –  plusepsilon.de Nov 13 '11 at 19:02

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up vote 1 down vote accepted

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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I will check the reference. Something similar was also my guess. Thanks –  plusepsilon.de Nov 14 '11 at 10:20

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