I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed intervals. I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall someone calling the notation with $[a,b]$ and $]a,b[$ as French notation.
What are the origins of the two notations?
Is the name French notation correct? Are they used frequently in France? Or were they perhaps prevalent in French mathematical community at some point? (In this MO answer Bourbaki is mentioned in connection with the notation $]a,b[$.)
Since several answerers have already mentioned that they have never seen $\langle a,b \rangle$ to be used for closed intervals, I have tried to look for some occurrences for this.
The best I can come up with is the article on Czech Wikipedia, where these notations are called Czech notation and French notation. Using $(a,b)$ and $[a,b]$ is called English notation in that article. (I am from Central Europe, too, so it is perhaps not that surprising that I have seen this notation in lectures.)
I also tried to google for interval langle
or "closed interval" langle. Surprisingly, this lead me to a question on MSE where this notation is used for open interval.
Best Answer
As a French student, all my math teachers (as well as the physics/biology/etc. ones) always used the $[a,b]$ and $]a,b[$ (and the "hybrid" $[a,b[$ and $]a,b]$) notations. We also, for integer intervals $\{a,a+1,...,b\}$, use the \llbracket\rrbracket notation (in LateX, package {stmaryrd}): $[[ a,b ]]$.
I have never seen the $\langle a,b \rangle$ notation used for intervals, though (only for inner products or more exotic binary operations).