Suppose that $X$ is topological vector space and let $A$ be a convex subset of $X$. Then can we say that $A$ is connected?
My Reasoning: Let $a_1,a_2\in A$. Then $ta_1+(1-t)a_2\in A$ for all $t\in [0,1]$, due to $A$ is convex. Now, since $X$ is a topological vector space, scalar multiplication is continuous in $X$. Therefore, $t\to ta_1+(1-t)a_2$ is a continuous map. Thus, the function $f:[0,1]\to X$, defined by $f(t)=ta_1+(1-t)a_2$ is a path from $a_1$ to $a_2$. So $A$ is path connected, hence connected.
In particular, any convex set in norm topology, weak topology, weak$^*$ topology is connected.
Any solution, comment and suggestion will be appreciated.
Yes, indeed, any convex subset in a TVS is path-connected (if the scalars include $\Bbb R$ of course) and hence connected. The argument is quite correct. Nitpick: your path $f$ goes from $a_2$ to $a_1$, not the other way around. Not really important, reverse the roles.