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This question already has an answer here:

I know that the $L^p$ spaces are reflexive for $1<p<\infty$. I want to explicitly show that $L^1((0,1),\mathbb{R})$ is not reflexive by finding an element of $L^1$** that is not in $L^1$. To be more precise:

There is a canonical embedding $J: X \rightarrow X^{**}$ that is given by

$$J(x)(y)=y(x)$$

I want to find an element of $L^1$** that doesn't get mapped to by $J$.

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marked as duplicate by Rhys Steele, Community Mar 4 at 16:41

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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This was answered in this question, the top two answers have a textbook reference and an actual proof.

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    $\begingroup$ If the question has already been answered on this site then the correct thing to do is to mark it as a duplicate (click on "close" and then "duplicate of") and not to post an answer with a link. $\endgroup$ – Rhys Steele Mar 4 at 16:41
  • $\begingroup$ Apologies, will do. $\endgroup$ – R_B Mar 4 at 16:42

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