Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology.

Question

Let X be a Banach space. Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology.

Could anybody give me some hints to start?

Answer 1

You have to show that for every weakly open subset $U$ of $X$ containing $x_0+y_0$, there exists weakly open subsets $V$, $W$ of $X$ containing respectively $x_0$ and $y_0$, such that if $x \in V$, $y \in W$, $x+y \in U$.

I suggest that you prove it when $U$ is some $\{y, \phi(y) < a\}$ where $a$ is a real number and $\phi \in X^*$. Then you can prove it when $U$ is a finite intersection of such subsets and from this address the general case.

Answer 2

You have to say what topology you are using on $X \times X$. Assuming that you are using the product topology here is a simple proof: if $(x_i,y_i) \to (x,y)$ weakly and $f$ is a continuous linear functional then $x_i \to x$ and $y_i \to y$ weakly by definition of product topology. Hence $f(x_i) \to f(x)$ and $f(y_i) \to f(y)$ and $f(x_i+y_i)=f(x_i)+f(y_i) \to f(x)+f(y)=f(x+y)$.

Remark: here $(x_i)$ and $(y_i)$ are nets. You can use sequences in the case of metrizable spaces but not in general.