收敛性与紧集列

To prove that the convergence in $\mathcal{E}(\Omega)$ is independent of the choice of the sequence of compact subsets $K_m$, we need to show that any two such sequences define equivalent families of seminorms, and thus the same topology on $\mathcal{E}(\Omega)$.

Let’s consider two sequences of compact subsets:

1. $\{K_m\}_{m=1}^\infty$, with $K_1 \subset K_2 \subset \cdots \subset \Omega$ and $\bigcup_{m=1}^\infty K_m = \Omega$.
2. $\{K_m'\}_{m=1}^\infty$, with $K_1' \subset K_2' \subset \cdots \subset \Omega$ and $\bigcup_{m=1}^\infty K_m' = \Omega$.

Define the seminorms associated with each sequence:

• For $\{K_m\}$: $\|\varphi\|_m = \sum_{|\alpha| \leq m} \max_{x \in K_m} |\partial^\alpha \varphi(x)|.$

• For $\{K_m'\}$: $\|\varphi\|_m' = \sum_{|\alpha| \leq m} \max_{x \in K_m'} |\partial^\alpha \varphi(x)|.$

Our goal is to show that these seminorms are equivalent, meaning that for each $m$, there exist constants $C_1, C_2 > 0$ and indices $m', m''$ such that:

1. $\|\varphi\|_m \leq C_1 \|\varphi\|_{m'}'$ for all $\varphi \in \mathcal{E}(\Omega)$.
2. $\|\varphi\|_{m'}' \leq C_2 \|\varphi\|_{m''}$ for all $\varphi \in \mathcal{E}(\Omega)$.

Proof:

1. For any fixed $m$, there exists $m'$ such that $K_m \subset K_{m'}'$:

   • Since $\bigcup_{k=1}^\infty K_k' = \Omega$ and $K_m$ is compact, it must be contained in some $K_{m'}'$.
   • Choose $m' \geq m$ so that $K_m \subset K_{m'}'$.

2. Relate the seminorms using the inclusion $K_m \subset K_{m'}'$:

   • For all $\varphi \in \mathcal{E}(\Omega)$: $\|\varphi\|_m = \sum_{|\alpha| \leq m} \max_{x \in K_m} |\partial^\alpha \varphi(x)| \leq \sum_{|\alpha| \leq m} \max_{x \in K_{m'}'} |\partial^\alpha \varphi(x)| \leq \|\varphi\|_{m'}'.$

3. Similarly, for any fixed $m'$, there exists $m''$ such that $K_{m'}' \subset K_{m''}$:

   • Since $\bigcup_{k=1}^\infty K_k = \Omega$ and $K_{m'}'$ is compact, it must be contained in some $K_{m''}$.
   • Choose $m'' \geq m'$ so that $K_{m'}' \subset K_{m''}$.

4. Relate the seminorms using the inclusion $K_{m'}' \subset K_{m''}$:

   • For all $\varphi \in \mathcal{E}(\Omega)$: $\|\varphi\|_{m'}' = \sum_{|\alpha| \leq m'} \max_{x \in K_{m'}'} |\partial^\alpha \varphi(x)| \leq \sum_{|\alpha| \leq m'} \max_{x \in K_{m''}} |\partial^\alpha \varphi(x)| \leq \|\varphi\|_{m''}.$

5. Combining the inequalities:

   • This shows that for appropriate indices $m', m''$, the seminorms satisfy: $\|\varphi\|_m \leq \|\varphi\|_{m'}' \leq \|\varphi\|_{m''}.$
   • Therefore, the seminorms $\|\cdot\|_m$ and $\|\cdot\|_m'$ are equivalent up to reindexing.

Conclusion:

Since the seminorms defined by different sequences of compact subsets are equivalent, they induce the same topology on $\mathcal{E}(\Omega)$. Consequently, convergence in $\mathcal{E}(\Omega)$ does not depend on the specific choice of the sequence $\{K_m\}$.

Answer: 

Because any two sequences of compact subsets increasing to Ω can be used to define equivalent families of seminorms, the convergence in $\mathcal{E}(\Omega)$ is independent of the choice of compact subsets $K_m$; thus, the different seminorms define the same topology.