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I'm reading a book about Functional Analysis, and now I've reached to the part about dual operators.

I'm having some difficulties understanding the following definition -

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Why $A^*$ is $Y^*\rightarrow X^*$? We know that $\phi \in Y^*$, i.e., $\phi:Y\rightarrow Y$ and bounded. So the image of $\phi(Ax)$ should be in $Y^*$, shouldn't it?

How come it's in $X^*$?

($X^*$ is reffered here as the space of all bounded linear functionals on $X$).

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First of all, $\phi$ is a functional, so it is $\phi:Y\to\mathbb{C}$.

Next, what is $A^*$? It takes a functional $\phi\in Y^*$ and returns a function $A^*(\phi)=f$ which is defined by $f(x)=\phi(A(x))$. If we take a vector $x\in X$ then $A(x)\in Y$ and then $\phi(A(x))\in\mathbb{C}$. So $f:X\to\mathbb{C}$. Then of course you need to check that $f$ is linear and bounded, but that's easy. So $f\in X^*$.

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