Definition of a weak-star weak-star continuous function

Question

I have seen the phrase "weak-star weak-star continuous" many times. But I'm don't know what it means for a function to be weak-star weak-star continuous. I just assumed it means that the function is continuous in the weak topology.

Could someone provide a formal definition and an example on this?

Thank you in advance!

Answer 1

Let $X,Y$ be Banach spaces, and $X^*, Y^*$ their conjugate spaces. Then each of the spaces $X^*$ and $Y^*$ as a weak* topology. A function $\phi : X^* \to Y^*$ is called weak* - weak* continuous if it is continuous according to these two weak* topologies.

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Let $X$ be a (real) Banach space, and $X^*$ the conjugate space. Of course $X^*$ is a set of functions $X \to \mathbb R$. The weak* topology for $X^*$ is the topology of pointwise convergence. That is: the weakest topology on $X^*$ such that the maps $f \mapsto f(x)$ are continuous for all $x \in X$.

Answer 2

Defining the continuity of a function in topological spaces requires two topologies, one on the domain and one on the codomain. It might be relatively easy to forget that fact if you use the same specific spaces and topologies every time like with $\mathbb{R}$, metric spaces or normed spaces in general, which is why often a simple "continuous" is unambiguous enough in a lot of situations, but not with weak topologies of course.

Now, the weak-$*$ topology on the (continuous) dual $X^*$ of $X$ a topological $\mathbb{K}$-vector space (it does not need to be normed as commented by Marco below, but this includes the case $X$ normed) ($\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$) is the topology $\tau_{w^*}(X) := \{\Phi_x^{-1}(U) \mid U \text{ open in }\mathbb{K},\, x \in X\}$ (personal notations, probably not standard), where $\Phi_x$ is the evaluation map $\Phi_x : \Lambda \in X^* \mapsto \Lambda(x) \in \mathbb{K}$.

Hence, a map $f : X^* \to Y^*$ is said to be weak-$*$ weak-$*$ continuous if it is continuous between $(X^*, \tau_{w^*}(X))$ and $(Y^*, \tau_{w^*}(Y))$, that is if for all $y \in Y$ and $V$ open in $\mathbb{K}$ there exists some $x \in X$ and some $U$ open in $\mathbb{K}$ such that $f^{-1}(\Phi_y^{-1}(V)) = \Phi_x^{-1}(U)$.

Of course, it is relatively close in concept to standard weak topologies, where the weak topology of a TVS $Z$ would be $\{\Lambda^{-1}(U) \mid U \text{ open in }\mathbb{K},\, \Lambda \in Z^*\}$, but of course for infinite-dimensional spaces the weak topology on $X^*$ and the weak-$*$ topology on $X^*$ are different (at least for normed spaces), since the key difference lies in the weak topology summoning all the functionals from $X^{**}$ while the weak-$*$ one is only concerned with the evaluation maps!

For more info on weak and weak-$*$ topology I can recommend Rudin's Functional Analysis and Conway's A Course on Functional Analysis.