How does one show any inner product space is connected? Should I start with assuming that it is not connected, and arrive at a contradiction? So let $X$ be an inner product space, and there exist open, disjoint, nonempty $U$ and $V$ so that $U\cup V = X$.
Or should I start with an equivalent definition of connectedness: the only clopen subsets of $X$ are $X$ and $\emptyset$? Assume $X$ is not connected, then there exists a clopen $U$ in $X$...
Honestly I must say I am completely lost with this. I'm not sure where to begin. In other words, I fail to see the significance of our space $X$ being an inner product space. If someone could give me a starting point I would really appreciate it!
An inner product space is a vector space, so for any $x,y\in V$ just check that the path $$ t\mapsto tx+(1-t)y $$ is continuous from $[0,1]$ to $V$. This will prove path-connectedness.