what exactly is the defintion of Y-weak topology?

Question

Recently, I read a textbook by Barry Simon,and it is opertor theory, a comprehensive course in analysis part 4. In the Banach space notation part, he says that: For $X$ be banach space, $\sigma(X,Y)$=$Y$-weak topology($Y$ are linear functional acting on $X$, $\sigma(X,Y)$ is the weakest topology on X in which $x\mapsto \langle y,x\rangle$ is continuous for each $y$. The question is: I am not sure whether we "definitely" need to restrict the topology in the range of the map (ie. the range of this map $x\mapsto \langle y,x\rangle$, we usually suppose to be $\Bbb R$ or $\Bbb C$) to be standard topology, or any kind of topology on the range is fine? Does this is the defintion what he want, or in fact, any kind of topology is ok?

Answer 1

After a few days thinking, I find that I am not focusing what the truely motivation means. I mean, this weak topology will definitely dependen on the topology on the range of the map(we can even find any absurd topology), but it doesn't matter. We only want to the map is continous, and the topology smallest, which is the key point. I think he say this definiton just for convenience for doing researching, and generalize to more abstract space.