1
$\begingroup$

Let K be a compact set. How does one show the following?

If a linear map $T:C^\infty_c(K) \to X$ into a normed vector space X is continuous then there exists $k \geq 0$ and $C>0$ such that $\|Tf\|_X \leq C\|f\|_{C^k}$ for all $f \in C^\infty_c(K)$.

$C^\infty_c(K)$ is not a normed vector space, correct? What notions of continuity hold for a map $C^\infty_c(K) \to X$, just the notion for a map between two topological spaces?

  • 0
    @MartinArgerami: The topology on $C^\infty_c(K)$, where $K \subset \mathbb{R}^d$, is generated by the family of norms $\|f\|_{C^k}:=\sup_{x\in \mathbb{R}^d} \sum_{j=0}^k|\nabla^j f(x)|$, for $k=0,1,\cdots$. This gives them the structure of Frech\'et spaces. A sequence $f_n \in C^\infty_c(K)$ converges to a limit $f\in C^\infty_c(K)$ iff $\nabla^jf_n$ converges uniformly on $K$ to $\nabla^j f$ for all $j=0,1\cdots$.2012-11-14

0 Answers 0