So in the text Functional Analysis Walter Rudin says that
"A topological vector space is a real or complex vector space equipped with a $T_1$-topology with respect to which the vector operations are continuous."
Now for a real t.v.s. $X$ it seem to me that the function $$ \gamma:[0,1]\ni t\longrightarrow x_1+t\cdot(x_2-x_1)\in X $$ for any $x_1,x_2\in X$ is continuous: indeed the vector sum $s:X\times X\longrightarrow X$ and the scalar-vector product $p:\Bbb R\times X\longrightarrow X$ are continuous and moreover even the functions $$ \varphi:[0,1]\ni t\longrightarrow(t,x_2-x_1)\in\Bbb R\times X\quad\text{and}\quad\psi:X\ni x\longrightarrow(x_1,x)\in X\times X $$ is continuous with respect product topology and thus observing that $$ \gamma=s\circ\psi\circ p\circ\varphi $$ we conclude that $\gamma$ is continuous. So if $\gamma$ is continuous then $X$ is path connected and so connected and I think that this is a really relevant result but unfortunately any author (even Rudin) does not observe this so that I thought to put a specific question where I ask clarification about. Morover I would like to know if also a complex t.v.s. is path connected. So could somoeone help me, please?