weird brackets in unit interval

Question

I have found during looking at the book Linear Algebra and its Applications (K. Nordstrom) some weird (for me) notation for belonging to the unit interval, namely, $\lambda \in \ ] 0,1 [$. Does it mean as always that $\lambda$ belongs to $[0,1]$ or something different?

Answer 1

The notation $]a,b[$ is used for an open interval, more commonly written as $(a,b)$; meaning: $$x \in \; ]a,b[ \; \iff a \color{red}{<} x \color{red}{<} b$$ whereas: $$x \in [a,b] \iff a \color{blue}{\le} x \color{blue}{\le} b$$ So $\lambda \in \; ]0,1[$ would mean values satisfying $0<\lambda<1$, excluding the end points of the interval.

Half-open intervals are then written in a similar way, e.g. $[a,b) = [a,b[$ etc.

This notation is more common in the French school (and countries adopting that notation) and has the advantage of avoiding confusion since $(a,b)$ is a common notation with other meanings too.

Answer 2

The interval $]0,1[$ is the same as $(0,1)$. Further $]0,1]=(0,1]$ and $[0,1[=[0,1)$. It is just an other notation.

If you use the inverse bracket, you can avoid misunderstanding $(0,1)$ as an element of $\mathbb R^2$ instead of the open interval.