$$+:X\times X\to X,\\(x,y)\mapsto +(x,y)=x+y$$ and $$\cdot:\Bbb{R}\times X\to X,\\(x,y)\mapsto \cdot(\lambda,y)=\lambda\cdot x$$ are weakly continuous, where $X$ is an infinite dimensional normed linear space.
My trial
Define for $\;i=1,2,$ \begin{align}\phi_i:(X,&\omega)\to (X\times X,\tau_X\times \tau_X),\\&x\mapsto \phi_i(x)=(x,y) \end{align} where $\omega$ is the weak topology on $E$. By definition of product topology, $\phi_i$ for $\;i=1,2,$ is continuous. So, \begin{align}+\circ \phi_i:(X,&\omega)\to (X,\tau_X)\\&x\mapsto x+y \end{align} is weakly continuous.
Similarly, for $\;i=1,2,$ define \begin{align}\phi_i:(X,&\omega)\to (\Bbb{R}\times X,|\cdot|\times \tau_X),\\&x\mapsto \phi_i(x)=(x,y). \end{align} By definition of product topology, $\phi_i$ for $\;i=1,2,$ is continuous. So, \begin{align}\cdot\circ \phi_i:(X,&\omega)\to (X,\tau_X)\\&x\mapsto \lambda\cdot x \end{align} is weakly continuous.
Please, I'm I right? If yes, can you please explain it to me clearly? If I'm wrong, can you please, provide another proof? I'm new to weak topology.