When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms of the inverse image of open subsets of the base field $ \mathbb{K} $ under continuous linear functionals.

My questions are:

1. Are the two definitions equivalent, and if so, how? If they are not equivalent, then what is the common definition of the weak topology?
2. For the first definition (in terms of semi-norms), how can we define the open sets if we do not have a single fixed semi-norm?
3. For the second definition, by ‘open subsets of $ \mathbb{K} $’, do we mean the open sets belonging to the usual topology on $ \mathbb{K} $?
4. The topological dual space of $ X $ is defined to be the set of continuous linear functionals defined on $ X $. Is this continuity with respect to each topology or just a fixed one?

I am so confused. Could someone help me to understand the definition of the weak topology? Thanks!