# The dual of $C_0^k(X)$

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

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).