I am trying to understand the statement of Theorem 7.7.1 in The analysis of linear partial differential operator I by Hormander. What exactly is meant by $f$ stays in a bounded set in $C^{k}(X)$? I assume it is with respect to some norm on this space but it isn't lear to me what the norm is. Any clarification appreciated. Thank you.
Notation. $X$ is an open subset of $\mathbb{R}^n$ and $C^{k}(X)$ is the space of real valued functions defined on $X$ which are $k$ times continuously differentiable.
Theorem 7.7.1 asserts that if $B \subset C^{k+1}(X)$ is bounded, then there exists a constant $C>0$ such that the claimed estimate holds for all $u \in C_0^k(K)$ and $f \in B$ such that $\operatorname{Im} f \geq 0$ in $X.$
What does it mean for $B$ to be bounded? It appears as though Hörmander does not explicitly define this, but the space $C^k(X)$ is equipped with the seminorms $\lVert \cdot \rVert_{C^k(K)}$ for all $K \subset X$ compact, which gives it the structure of a locally convex space. We say $B$ is bounded if it is bounded with respect to all of these seminorms, that is $\sup_{f\in B} \lVert f \rVert_{C^k(K)} < \infty$ for all $K \subset X$ compact (see Definition 9 here).