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Let $E$ be a normed space and $F$ a subspace of $E$. Is there any linear mapping $T: F' \to E'$ such that $||T(\phi)|| = ||\phi||$ for every $\phi \in F$?

Here is what I am thiking. Take $p < q$, Hölder conjugates. Then, there is the natural inclusion $\ell^p \hookrightarrow \ell^q$. If the previous question were to have a positive answer, then we would have a linear, isometric inclusion $\ell^q \hookrightarrow \ell^p$, and I do not know if that is possible.

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  • No. Look at Pitt's theorem. – Jochen Jun 08 '18 at 07:43
  • To expand on what Jochen is saying Pitt's Theorem implies that any bounded operator from $l^{q}$ into $l^{p}$ with $q>p$ is compact. If an isometry is compact then the domain is finite dimensional because its unit ball is compact. This is a contradiction. Reference for Pitt's Theorem: https://math.stackexchange.com/questions/218284/proof-of-pitts-theorem – Kavi Rama Murthy Jun 08 '18 at 08:56

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