What does "total in L2" mean?

Question

I cannot figure out what "total in $L^2$" means. The following is the complete sentence:

Prove that $\{\chi_{(a,x)}:~x\in[a,b]\}$ is total in $L^2([a,b],dx)$ where chi's are the characteristic functions of given intervals.

Answer 1

Linear Operators in Hilbert Spaces, Joachim Weidmann, page 35:

Given a subspace $T$ of $H$, $M$ is total in $T$ if $M\subseteq T$ and $T\subseteq\overline{L(M)}$.

Here $H$ is a Hilbert space (pre-Hilbert space is allowed), and $L(M)$ is the span of elements of $M$.

Answer 2

The meaning of total is as given by user284331. For the proof the equivalent definition: a set of vectors in a Hilbert space is total iff the only element orthogonal to each of these vectors is the zero vector. Suppose $f \in L^{2}$ and f is orthogonal to $\chi_{(a,x)}$ for each x. Then $\int _x ^{y} f(t)dt =0$ whenever $a \leq x <y \leq b$. One way of showing that $f=0$ almost everywhere is to use Lebesgue's Theorem on differentiation of indefinite integrals.Another approach is to use the approximation theorem of measure theory.