Duals of embeddings in the space of distributions

Question

If $ \Lambda \colon X \hookrightarrow \mathcal{D}'$ is a continuous embedding of a normed vector space $X$ into the space of distributions (for example $X=L^p$), is it true that the dual of $\Lambda(X)$ (equipped with the topology of $\mathcal{D}'$) is the same as the dual of $X$?

Answer 1

Before getting into distributions, consider a simpler example: $X=L^2[0,1]$ continuously embeds into $Y=L^1[0,1]$ via the identity map $\Lambda f=f$. The dual of $\Lambda(X)$ is $Y^* = L^\infty[0,1]$, since $\Lambda(X)$ is dense in $Y$. This is not the same as $X^*$.

On the positive note, the adjoint operator $\Lambda^* : Y^*\to X^*$ is a continuous embedding. But it is not surjective.

Same thing with distributions. The dual of $L^p$ is $L^q$. But a general $L^q$ function cannot act on distributions. E.g., it cannot act on Dirac delta. So, the set of continuous linear functions on $\Lambda(X)\subset \mathcal D'$ is smaller that $X^*$.