Prove that $\left\{\mathcal{X}(a,x):x \in [a,b]\right\}$ is total in $L^2([a, b]), dx)$ where $\mathcal{X}(c,d)$ is the characteristic function of $(c, d)$.
What is the definition of being total? This problem is from the book Barry Simon. And that definition does not appear. Does it mean to be dense?
"Total" means that the linear span of your set (i.e., the set of all functions of the form $c_1 \mathcal{X}(a,x_1) + c_2 \mathcal{X}(a, x_2) + \dots + c_n \mathcal{X}(a, x_n)$) is dense in $L^2([a,b])$.
Equivalently, the only function which is orthogonal to every function in your set, is the zero function. (Proving these are equivalent is a good exercise.)