# Inner product space is connected

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.
• Ahh, so $\gamma(t)% := tx+(1−t)y$ will be a map connecting any two vectors in $V$ and $\gamma(0) = y$, $\gamma(1) = x$, and I just show $\gamma$ is continuous and so it's path connected, and of course is is well-known that path-connectedness $\Rightarrow$ connectedness. Thanks for that. Commented Apr 20, 2014 at 7:31