# Literature on relationship between strong and weak* (weak star) convergence

I am trying to follow a proof in the paper Wasserstein Generative Adversarial Networks by Arjovsky et al. (proof A in supplementary material).

They show that the convergenc of the total variation distance of two probability distributions corresponds to convergence in a strong topology induced by the norm and the convergence of the Wasserstein-1 distance of two distributions corresponds to convergence in a weak* topology.

If I understand that correctly, the convergence of a sequence in a topology induced by the norm always converges strongly (see here). And convergence in a weak* topology is weak* convergence by definition.

As far as I understand this, strong convergence induces weak* convergence. I did also find that in some lecture notes (e. g. here or here), but not in any textbook or scientific article. So does anyone know of literature I could cite in my thesis concerning this?

Or a proof that strong convergence induces weak* convergence (easily) understandable by electrical engineers would be fine, too, I guess.

Thank you very much in advance!

welcome to MSE.

The fact that strong convergence implies weak*-convergence in the dual space is so basic, that most books don't bother mentioning it. What might be a good thing to do is to reference "Functional Analysis" 2nd edition by Walter Rudin which is a good book on functional analysis containing a good introduction to the weak and weak* topology.

As for the proof, this consists of two parts, showing that strong convergence implies weak convergence and noting that weak convergence implies weak*-convergence.

Let $$X$$ be a normed vector space and let $$X^{*}$$ denote its dual space. By definition $$X^{*}$$ is the space of all continuous linear functional, so for any $$\varphi\in X^{*}$$ the operator norm $$\|\varphi\|=\sup\{|\varphi(x)|:x\in X,\|x\|\leq1\}$$ is finite. Now suppose $$(x_{\alpha})$$ is a net in $$X$$ (or $$(x_{n})$$ is a sequence if you don't want to use nets) converging strongly to $$x\in X$$. Then for all $$\varphi\in X^{*}$$ we find $$\lim_{\alpha}|\varphi(x_{\alpha})-\varphi(x)|=\lim_{\alpha}|\varphi(x_{\alpha}-x)|\leq\lim_{\alpha}\|\varphi\|\|x_{\alpha}-x\|=0.$$ As this holds for all $$\varphi\in X^{*}$$ by definition $$(x_{\alpha})$$ weakly converges to $$x$$.

For the next part it is important to note that the weak* topology is not defined on $$X$$ but $$X^{*}$$. So we are considering strong convergence on $$X^{*}$$ with respect to the operator norm, and a net $$(\varphi_{\alpha})$$ in $$X^{*}$$ weakly converges to $$\varphi$$ if for all $$\psi\in (X^{*})^{*}$$ we have $$\lim_{\alpha}\psi(\varphi_{\alpha})=\psi(\varphi)$$. Recall that $$(\varphi_{\alpha})$$ converges to $$\varphi$$ in the weak* topology if for all $$x\in X$$ we have $$\lim_{\alpha}\varphi_{\alpha}(x)=\varphi(x)$$. Note however that for $$x\in X$$ the evaluation map $$\psi_{x}:X^{*}\rightarrow\mathbb{R},\varphi\mapsto\varphi(x)$$ is a continuous linear functional on $$X^{*}$$ hence $$\psi_{x}\in(X^{*})^{*}$$. Therefore if $$(\varphi_{\alpha})$$ converges to $$\varphi$$ in the weak topology it also converges in the weak* topology. (For a mathematical audience I would just write that the weak convergence implying weak* convergence is trivial.)

Combining these two results gives that strong convergence on $$X^{*}$$ implies weak* convergence.

Let me know if any of these steps is still unclear.

As far as I understand this, strong convergence induces weak$$^*$$ convergence... does anyone know of literature I could cite in my thesis concerning this?

Yes: Proposition 3.13(ii) in Brezis book.