Weak continuity of the addition and scalar multiplication

Question

Let $X$ be an infinite dimensional normed vector space. Show that vector addition and scalar multiplication are weakly continuous. $$+:X×X \rightarrow X; +(x,y)=x+y$$ $$•:\mathbb {R}×X \rightarrow X; •(\lambda,x)=\lambda x$$ My Ideas (From Brezis book):

1. A map $f$ is weakly continuous iff $\forall g \in X^*, g\circ f $ is continuous.

2. Strong to strong continuity of a linear map implies weak to weak continuity.

Can I conclude directly from 2. that since both $+$ and $•$ are strong to strong continuous, the result follows. Please I need hints on how to put the thought together

Answer 1

Weak - weak continuity of $(\lambda, x) \to \lambda x$: if $(\lambda_i,x_i) \to (\lambda,x)$ then $\lambda_i\to \lambda$ and $x_i \to x$ weakly and, for any $x^{*} \in X^{*}$, we have $x^{*} (\lambda_i x_i)=\lambda_i x^{*}(x_i) \to \lambda x^{*}(x)=x^{*} (\lambda x)$.

You have to note two points: a) there is only one norm topology on $\mathbb R$; weak and strong coincide here. b) Weak topology is not metrizable in infinite dimensional spaces so we have to use nets instead of sequences.

Answer 2

Yes, you are right, from 2. you will get the weak continuity of $+$ and $•$.