Definition of weakly continuous map from one Banach space to another

Question

What is the definition of a weakly continuous function from a Banach space to a Banach space?

Suppose $X$ and $Y$ are Banach spaces. Define $f : X \rightarrow Y$ as a function. Am I right to say that $f$ is weakly continuous if the net $x_{\alpha} \rightarrow x$ in the weak topology of $X$, then $f(x_\alpha) \rightarrow f(x)$ in the weak topology of $Y$?

I know that we cannot change the net into a sequence, as some Banach spaces are Schur spaces, which have the property that every weakly convergent is norm convergent.

If my definition is not correct, then may I know what is the proper definition?

Remark: Sorry for the confusion. I am talking about weak-to-weak continuous.

Answer 1

Both topologies matter: one on the domain, the other on the target space. Thus, one can speak of

• weak-weak continuity: the preimage of a weakly open set in $Y$ is weakly open in $X$.
• weak-strong continuity: the preimage of an open set in $Y$ is weakly open in $X$.
• strong-weak continuity: the preimage of a weakly open set in $Y$ is open in $X$.

The first one is often used: e.g., every continuous linear operator is weak-weak continuous. But so is the second: finite rank operators are weak-strong continuous. Also, compact linear operator is weak-strong continuous on bounded sets. I don't remember ever reading about strong-weak continuity, it's included just for completeness.

If you simply say "weakly continuous from $X$ to $Y$", the reader will have to pause and guess your intention. Try to make it clear from context which version you are using.