Operator topologies and examples In class we covered several operator topologies: the weak topology,  the weak* topology,  the weak operator topology, and the strong operator topology.  The first two are defined on a normed vector space $V$ and its dual $V$* resp., and the latter two are defined on $L(X,Y)$ for Banach spaces $X$ and $Y$.
However, I wasn't in class when we covered this so I'd like to ask: what motivates the above topologies, and are there any (relatively elementary) examples?  The relevant section in Folland does not say much outside of describing convergence and Alaoglu's theorem.
 A: The weak* topology is the topology of pointwise convergence: $f_n \to f$ in the weak* topology if and only if $f_n(x) \to f(x)$, which I think is a pretty good reason to care about it.
The weak topology on a topological vector space $V$ gives the convergence condition $v_n \to v$ if and only if for all $\lambda \in V^*$ we have $\lambda(v_n) \to \lambda(v)$. So you can see this is the same as the the weak* topology if $V$ is reflexive, meaning $(V^*)^* \cong V$, a condition that's satisfied by all finite-dimensional vector spaces: in this case the weak* topology on $V = (V^*)^*$ is given by $v_n \to v$ if and only if for all $\lambda \in V^*$ we have $\lambda(v_n) \to \lambda(v)$, again. So you can see the weak topology as a different generalization of the topology of pointwise convergence. 
Now why would you want to use the weak topology? Let $V$ be the space of $L^2$ real-valued functions on $[0,1]$, say. Then integration with respect to Lebesgue measure, or some other Radon measure $\mu$, say, is an element of $V^*$, and given that you care about when $\int f_n \, d\mu \to \int f \, d\mu$, the weak topology seems like a reasonable thing to care about.
Now for the strong operator topology. This is the topology of pointwise convergence, again. The balls about the origin in this topology are of the form $\{T \in L(X,Y) : |Tx| < \epsilon\}$ for fixed $x \in X$ and $\epsilon > 0$. This means two operators are close (in the intersection of many small balls) if they take nearby values on many points, so you get $T_n \to T$ if and only if $T_n x \to Tx$ for all $x \in X$, again.
The weak operator topology, on the other hand, is harder to motivate, if I recall. I think its value only becomes obvious once one starts to prove things, and sees that convergence in the weak operator topology is a convenient thing to ask. I am probably missing some obvious examples, as I last studied functional analysis years ago, but I hope this helps a bit.
