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I know that the space of all the bounded linear maps between two Banach spaces, denoted by $L(X,Y)$, has a relationship with the projective tensor space of $X$ and $Y$, $$({X \widehat\otimes_{\pi} Y})^* = L(X,Y^*).$$ Is there any relationship between the space of all compacts operators between the two spaces, denoted by $K(X,Y)$, and the projective tensor space of $X$ and $Y$?

Thanks in advance.

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    $\begingroup$ In the case of the Hilbert space $X=Y=\ell_2$, the space of compact operators on $X$ is a pre-dual of $X\otimes X$. The trace of these dualities on the space of diagonal operators is (isometrically) $c_0^*=\ell_1$ and $\ell_1^*=\ell_\infty$. $\endgroup$
    – Pietro Majer
    Commented Jun 9, 2021 at 5:19
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    $\begingroup$ The duality $(X\otimes_\varepsilon Y)^* \cong X^* \otimes_\pi Y^*$ holds true, more generally, if one of the duals has the approximation property and one of the duals has the Radon-Nikodym property. $\endgroup$
    – Dirk Werner
    Commented Jun 9, 2021 at 16:37
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    $\begingroup$ Ryan's book is a good source for starters link.springer.com/book/10.1007/978-1-4471-3903-4. $\endgroup$
    – Onur Oktay
    Commented Jun 14, 2021 at 16:15

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