TITLE: Two terms that I want to understand: weakest topology and jointly continuous (in the following context).
QUESTION [8 upvotes]: I was reading an article online, please help me to understand the following lines (in bold letters). - 
Topological structure:
If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and therefore a topology on V.the distance between two vectors u and v is given by ‖u−v‖. 
This topology is precisely the weakest topology which makes ‖·‖ continuous and which is compatible with the linear structure of V in the following sense:
1.The vector addition + : V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality.
2.The scalar multiplication · : K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.

Please explain me the Weak topology and how does it makes norm ‖·‖ continuous. what does it mean by "Addition + : V × V → V is jointly continuous with respect to this topology"
Thank you so much in advance.

REPLY [3 votes]: There is already a very nice answer, but I thought I could give an example of a function which is discontinuous as a function of two variables, but continuous in each variable. Consider the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$,
$$
f(x,y) = \begin{cases} 
      \frac{xy}{x^2 + y^2} & (x,y) \neq (0,0) \\
      0 & (x,y) = (0,0) \\
   \end{cases}
$$
This is discontinuous as it is 1/2 arbitrarily close to $(0,0)$ when $x=y$, but it is easily seen to be continuous in each variable.