# How is the weak-star topology useful?

Today I learnt something about the weak-star topology, but I don't know what the use of weak-star topology is. I hope someone can tell me what we can do with the weak-star topology. Thanks in advance!

• We have a nice compactness result, for starters: en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem – Zach L. Jun 4 '13 at 3:22
• Also, forget about what it's useful for at the moment, it's only natural to consider the topology of pointwise convergence for elements of the dual. On top of that, as Zack L mentioned, there are compactness results, such as Alaoglu and Krein-Milman. – Michael Jun 4 '13 at 3:26
• thanks. I have one more question. In class, the teacher said that the dual space of $L_{p}$ is $L_{q}$ while $1/p+1/q=1$, while when p is infinte there is an exception. We know that the dual of a space is the set of all the bounded fuctionals on it. How can we say that the dual space of $L_{p}$ is $L_{q}$ ,since, in my opinion, $L_{q}$ is not a space containing the functionals of $L_{p}$. – Brain Zhang Jun 4 '13 at 3:30
• @Martin: You're absolutely right. People will often say things are "equal" or "identical" when they mean "canonically isomorphic". Depending on how comfortable you are with this sloppiness, it can either a convenience or an annoyance. – Zev Chonoles Jun 4 '13 at 3:52
• Usually, if you think something your teacher said is "wrong", you will learn a lot by discussing it with her/him. – GEdgar Jun 4 '13 at 11:59

The main use of weak* topology is to provide a topolgy on $V^*$ for a normed space(or a TVS) $V$ such that the unit ball in $V^*$ is compact, which is the Banach-Alaoglu Theorem. There are numerous places in analysis where we use this topology, for example from Riesz Representation theorem, for any locally compact Hausdorff space $X$, if $\mathcal{M}(X)$ is the space of complex Radon measures, then the dual $C_0(X)^* \cong \mathcal{M}(X)$, So from usual sup norm topology of $C_0(X)$ you can induce the weak* topology on Radon measures, thus you can have convergence, compactness results in measures. If you are familiar with distribution theory, you find that given the inductive limit topology on $C^\infty_c(\Omega)$, weak* topology is induced on the space of distributions. This is a fundamental requirement in PDEs.