I know that the space of the all 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.

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.

I know that the space of the all 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.

I know that the space of the all 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.

I know that the set of the all linear applications between two spaces, denoted by L(X,Y) have a relationship with the tensor proyective space of X and Y, $$(\hat{X \oplus_{\pi} Y})^* = L(X,Y^*)$$. There is any relationship between the set of the all compacts operators between two spaces, denoted by K(X,Y) and the tensor proyective space of X and Y.

Thanks in advance.

I know that the space of the all 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.