Let


* $E$ be a $\mathbb R$-Banach space
* $E\:\hat\otimes\_\pi\:E$ denote the projective tensor product



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> How can we show that $E\:\hat\otimes\_\pi\:E$ is isomorphic to a subspace of $\left(E'\:\hat\otimes\_\pi\:E'\right)'$?
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Clearly, if $\mathfrak B(E'\times E')$ denotes the space of bounded bilinear forms on $E'\times E'$, then $\mathfrak B(E'\times E')$ is isometrically isomorphic to $\left(E'\:\hat\otimes\_\pi\:E'\right)'$.


So, we could conclude, if we would be able to show that $E\:\hat\otimes\_\pi\:E$ can be embedded into $\mathfrak B(E'\times E')$.


Actually, I know that $E\otimes E$ (the algebraic tensor product) can be embedded into $\mathcal B(E^\ast\times E^\ast)$ (the set of bilinear forms on the cartesian product of the algebraic dual space $E^\ast$ with itself).


A canonical choice for this embedding would be $$\sum\_{i=1}^nx\_i\otimes y\_i\mapsto\left((\varphi,\psi)\mapsto\sum\_{i=1}^n\varphi(x\_i)\psi(y\_i)\right)\tag1\;.$$ If $\iota$ denotes this embedding, then it's easy to see that $\iota$ is a bounded linear operator from $E\otimes\_\pi E$ (the algebraic tensor product $E\otimes E$ equipped with the projective norm) to $\mathfrak B(E'\times E')$ and hence admits a unique extension to a bounded linear operator $\overline\iota$ from $E\:\hat\otimes\_\pi\:E$ to $\mathfrak B(E'\times E')$.



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> If this approach is sensible at all, the only thing I need to conclude is the injectivity of $\overline\iota$. How can we show that?
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How can we show that $E\:\hat\otimes_\pi\:E$ is isomorphic to a subspace of $\left(E'\:\hat\otimes_\pi\:E'\right)'$?
As suspected by Yemon Choi the question is very closely related to the *approximation property*: As can be seen e.g. in the book *Tensor Norms and Operator Ideals* of Defant and Floret (page 64 combined with the remark 5.4) a Banach space $E$ has the approximation property if and only if the canonical mapping $E\tilde{\otimes}\_\pi F \to (E' \otimes\_\pi F')'$ is injective for all Banach spaces $F$ (or only for $F=E'$). I don't know if it is written somewhere but I would be very surprised if the mapping would always be injective for $F=E$.


The question as stated (whether $E\tilde{\otimes}\_\pi E$ is isomorphic to *some* subspace of $(E' \otimes\_\pi E')'$ in a possibly non-canonical way) is, of course, very different.


