Examples for $Y' \not\hookrightarrow X$ if $X$ is not a dense subset of $Y$

Let $X,Y$ normed $\mathbb{R}$-vector spaces. If $X$ is a dense subset of $Y$ and there is a continuous embedding $X \hookrightarrow Y$ then there holds: $Y'=\mathcal{L}(Y,\mathbb{R}) \hookrightarrow \mathcal{L}(X,\mathbb{R}) = X'$. I have just proven this claim.

Now, is there a simple example for $Y' \not\hookrightarrow X'$ if $X$ is not dense in $Y$?

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2022-07-25 20:42:38
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