I want a reference of the following statement which I think is true. Let $X$ and $Y$ be Banach spaces with $X$ finite dimensional. Then $(X\otimes\_\epsilon Y)^\*$ is isometrically isomorphic to $(X^\*\otimes\_\pi Y^\*)$ and $(X\otimes\_\pi Y)^\*$ is isometrically isomorphic to $(X\otimes\_\epsilon Y)^\*$ where $\otimes\_\epsilon$ and $\otimes\_\pi $ denote the injective and projective tensor product of Banach spaces respectively. For tensor product of Banach spaces see the book <https://www.amazon.in/Raymond-A-Ryan/e/B001K88BH0/ref=dp_byline_cont_pop_ebooks_1>.



Duality of projective and injective tensor product
Since Ryan's book is mentioned in the original question, here's some pointers for an answer based on this source:


* See Section 3.4 for what the dual space of $X\otimes\_\epsilon Y$ is. If you combine Proposition 3.14 (and the comments after) with Proposition 3.22 (see the comments after it) we get $(X\otimes\_\epsilon Y)^\* = I(X,Y^\*) = I(Y,X^\*)$ the space of integral operators.
* Now look at Theorem 5.33 which shows that when $X^\*$ has the Radon-Nikodym Property and the approximation property (which is true when $X$ is finite-dimensional) then $I(Y,X^\*) = X^\* \hat\otimes\_\pi Y^\*$. See also the proof of Proposition 5.52.


For the dual of $X\otimes\_\pi Y$ see Section 2.2 where it's shown that $(X\otimes\_\pi Y)^\* = B(X,Y^\*)$ the space of all bounded linear maps. We always have that $X^\* \hat\otimes\_\epsilon Y^\*$ is a subspace of $B(X,Y^\*)$, see Section 3.1, and it's easy to see that when $X$ is finite-dimensional, then we have equality here.


