If $X$ is a topological vector space then is a set $U$ open when the inner product of its element is positive?

Question

So let be $X$ a (real or complex) topological vector space and thus we suppose that the inner product of any pair of elements of $U$ is (strictly) positive, that is $$ \langle x,y\rangle>0 $$ for any $x,y\in U$. So I ask to clarify if $U$ is open provided that the vector topology of $X$ is just the vector topology generated by the inner product. So could someone help me, please?

Answer 1

So if $X=\Bbb R$ and $U:=[\epsilon, 1]$ for any $\epsilon>0$ then $$ \langle x,y\rangle=xy>0 $$ for any $x,y\in U$ but unfortunately $U$ is not open so that the statement is false.