Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C \left \| x \right \|_X$
for some $C \geq 0$ (general definition of continuous inclusions in normed spaces)

My problem is to show that $\mathcal{D}_K(\Omega) \hookrightarrow \mathcal{E}(\Omega)$ where there are the following notations:

The space of test functions is $\mathcal{D}(\Omega):=\cup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ where $K \subset \Omega$ is an arbitrary compact and obviously $\Omega \subseteq \mathbb{R}^n$ is an open nonempty.
Moreover $\mathcal{D}_K(\Omega):=\lbrace \varphi \in \mathcal{E}(\Omega):$ supp$(\varphi) \subset K\rbrace$ where $\mathcal{E}(\Omega)$ represent the locally convex space of all $C^\infty$ functions with the topology defined by increasing sequence of semi-norms

$\displaystyle q_{j}(f)=\sum_{|\alpha| \leq j} \left \| D^\alpha f \right \|_{K_{j}}= \sum_{|\alpha| \leq j} \sup_{x \in K_j} |D^\alpha f|$.

Now, $\mathcal{E}(\Omega)$ and $\mathcal{D}_K(\Omega)$ are Frechét spaces and in particular is defined the Fréchet norm associated to the sequence of semi-norms. For example, and in the general case, for sequence of semi-norms $\lbrace p_n \rbrace$ the Frechet norm is $\left \| x \right \|:=\sum_{n=1}^\infty 2^{-n} p_n(x)/(1+p_n(x))$. The topology in $\mathcal{D}_K(\Omega)$ is defined by increasing sequence of semi-norms

$\displaystyle q_N(f)=\sum_{|\alpha| \leq N} \left \| D^\alpha f \right \|_{K_{N}} = \sum_{|\alpha| \leq N} \sup_{x \in K} |D^\alpha f|$

Since $\Omega$ is the union of an increasing sequence of the compact sets $\lbrace K_j \rbrace_{j \in \mathbb{N}}$, for any compact set $K=K_N \subset \Omega$ which corresponds to a fixed $N \in \mathbb{N}$, we have that $q_j(f) \leq q_N(f)$ for all $j \leq N$, since $K_j \subset K$. Therefore, in terms of Fréchet norm, follows the thesis.

However, I’m thinking that is most appropriate to define the continuous inclusion in locally convex spaces. I remembered also another result that extends the continuity condition for linear operators between normed spaces in locally convex spaces:

  1. “Let $\mathcal{P}$ and $\mathcal{Q}$ are two sufficient families (sufficient means that $p(x)=0$ for every $p \in \mathcal{P}$ implies $x=0$) of semi-norms for the locally convex space $E$ and $F$. A linear application $T: E \rightarrow F$ is continuous if and only if for every $q \in \mathcal{Q}$ there exist $p_j \in \mathcal{P}$ with $j \in J$ finite and a constant $C \geq 0$ so that $q(Tx) \leq C \max_{j \in J} p_j(x)$.”

In particular, if the locally convex spaces $E$, $F$ are normable, by definition we have $\mathcal{P}=\lbrace \left \| \cdot \right \|_E \rbrace$ and $\mathcal{Q}=\lbrace \left \| \cdot \right \|_F \rbrace$ and for the previous result we find the condition of continuity for linear operators in normed spaces.

Then for this result, we say that $E \hookrightarrow F$ if the inclusion $\iota : E \rightarrow F$ is continuous, i.e. for every $q \in \mathcal{Q}$ there exist $p_j \in \mathcal{P}$ so that $q(x) \leq C \max_{j \in J} p_j(x)$ for some $C \geq 0$.

Consequently, in the above procedure, it is unnecessary to consider the Fréchet norm but remains correct for (1). It seems to me better and more elegant.

it’s correct ?

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