On the definition of weak and weak-* topologies I have been studying topological vector spaces, and despite going over numerous resources, the definitions of weak and weak-* topologies have been causing me some confusion. I am having trouble visualizing and understanding these topologies.
Suppose $X$ is a normed vector space.
Then the weak topology on $X$ is the topology generated by $X^*$, in other words the weakest topology making $x \mapsto f(x)$ continuous for all $f \in X^*$.
Similarly, the weak-* topology is the weakest topology making the maps $x \mapsto f(x)$ continuous for all $x \in X$.
I see the big difference here is that one is generated by the dual and the other by the original vector space. However, I have three points of confusion. I suspect part of my difficulty may be due to not properly visualizing topologies generated by a collection of seminorms.

*

*The dual space $X^*$ is defined as the set of bounded linear functionals from $X$ to the underlying field. However, I recall reading that the boundedness of a linear map is equivalent to the map being continuous, so I fail to see what sets we are excluding in this new, weaker topology.


*What do the open sets (or more simply, the basis sets) look like in these two topologies?


*The resources I am learning this from often note a relation between the double dual $X^{**}$ and the weak-* topology, what is the relation between these spaces exactly?
 A: For any $X$, the weak topology on $X$ is defined to be the coarsest topology that makes $x\mapsto f(x)$ continuous for each $f\in X^*$, the dual space of $X$. Equivalently it is the topology induced by the seminorms $x\mapsto|f(x)|$, and an open neighborhood of origin looks like $\{x\in X: |f_i(x)|\leq\epsilon,\forall\ 1\leq i\leq n \}$. More generally, weak topology makes sense whenever we have a pairing $(X,Y)$ of spaces, see the wiki article.
Now this definition works for any normed space, so in particular it also works for $X^*$: the weak topology on $X^*$ is the coarsest topology that makes $f\mapsto \chi(f)$ continuous for each $\chi\in X^{**}$, the dual space of $X^*$. However there is another natural topology on $X^*$, where we only consider those $\chi$ that comes from elements in $X$, namely we have an embedding $i:X\rightarrow X^{**},x\mapsto\chi_x$ where $\chi_x(f):=f(x)$. This is the weak* topology. In general the weak* topology is weaker than the weak topology on $X^*$, but if $i(X)=X^{**}$ then obviously they are the same (the converse is also true, see here)
The weak/weak* topology is much weaker than the norm topology. Every weak open neighborhood of origin is norm-unbounded. The closed unit ball of $X^*$ is compact in weak* topology (Alaoglu Theorem), while it is far from compact under the norm topology, unless $X$ is finite dimensional.
Hilbert space is an important special case. It is reflexive by Riesz representation theorem, so no need to distinguish weak and weak* topology. Say $(e_i)_{i=1}^\infty$ is an orthonormal basis for the Hilbert space $H$, so each $x\in H$ can be expanded as $x=\sum x_ie_i$, also denoted as $(x_1,x_2,x_3,...)$. Then on any norm-bounded set, weak topology is exactly componentwise convergence, i.e., a sequence $x^{(n)}$ converges to $y$ iff $x^{(n)}_i$ converges to $y_i$ for every $i$. This is a good exercise, you can also find it in this note. There is also an operator version.
