I can't really wrap my head around the definitions of topologies on Banach spaces. My question concern what these definitions actually mean and if I understand them correctly.
We let $X,U$ be Banach spaces and $\mathcal{L}(X,U)$ denote the set of bounded linear maps from $X$ to $U$. In Peter D. Lax book on functional analysis, page 165 the following definitions are found:
The strong topology in $\mathcal{L}(X,U)$ is the weakest topology in which all functions $\mathcal{L}\rightarrow U$ of the form $\mathbf{M}\mapsto \mathbf{M}x$ are continuous, $x$ being any point of $X$.
The weak topology in $\mathcal{L}(X,U)$ is the weakest topology in which all linear functionals of the form $\mathbf{M}\rightarrow l(\mathbf{M}x)$ are continuous, $x$ being any point of $X$ and $l$ any point in $U'$ (the dual $U$).
Exercise 5. Define the weak* topology in $\mathcal{L}(X,U')$... Show that there is a natural correspondence one-to-onebetween $\mathcal{L}(X,U')$ and $\mathcal{L}(U,X')$, and that this correspondence is continuous in the weak* topology.
So I thought that subbasis elements for the topologies are given by the following:
STRONG TOPOLOGY:
We want the inverse image of open sets of $U$ to be open in $\mathcal{L}(X,U)$. Therefore subbasis elements are given by
$$W_{\mathbf{M}_0,x,\varepsilon} = \{\textbf{M}\in \mathcal {L}(X,U) \mid \|\mathbf{M}_0x-\mathbf{M}x\|_U<\varepsilon\}$$
where $x\in X$, $\varepsilon>0$ and $\mathbf{M}_0\in \mathcal{L}(X,U)$. Then inverse images of open balls are open and therefore also inverse images of open sets.
WEAK TOPOLOGY:
$$W_{\mathbf{M}_0,x,l,\varepsilon}=\{\mathbf{M}\in \mathcal{L}(X,U)\mid |l(\mathbf{M}x)-l(\mathbf{M}_0x)|<\varepsilon\}$$
where $x\in X$, $l\in U'$ and $\varepsilon>0$. Then again inverse images of open balls are again open.
WEAK* TOPOLOGY
For this one I am genuinely confused what this actually means and was wondering what the subbasis elements look like. Previously he has said that the weak* topology in a Banach space $U = X'$ is the crudest topology in which all linear functionals $l(x)$ for fix $x$ are continuous. This would be generated by
$$W_{l_0,x,\varepsilon} = \{l\in U\mid |l(x)-l_0(x)|<\varepsilon\}.$$
How could this reasoning be applied in this case? $\mathcal{L}(X,U')$ is not the dual of any space? My previous attempt at this definition can be found
What is the weak star topology on $L(X,U')$?
but I don't see what it has to do with weak* topologies.
I know this question is long but it could be summarised as: are the first two definitions correct and how do we define weak* topology in this specific case?