TITLE: Is Inner product continuous when one arg is fixed?
QUESTION [37 upvotes]: In a inner product space with inner product $\langle\ ,\ \rangle$ and real or complex line as its base field, for each point $x$ in the space, is $\langle x,-\rangle$ continuous function on the second argument, and is $\langle - ,x\rangle$ continuous function on the first argument? "Continuous" is defined respect to the topology induced by the inner product.
Thanks and regards!

REPLY [5 votes]: I think the easiest way is to show that it is a convex function, and then use the theorem that says that if a convex function defined on a convex set, then the function is continuous in every interior point. Since we are talking about $\Bbb{R}^n$, then it is true for all points because $\Bbb{R}^n$ is a convex and open set.