Let $$\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\},$$ where we take $$\eta_\alpha(t)= \left\{ \frac{\alpha}{t} \right\} -\alpha \left\{ \frac{1}{t} \right\},$$ and $ \left\{ x \right\} $ is the fractional part function.
Previous is a closed and convex subset of the Hilbert space $L^2(0,1)$.
This set $\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\}$ is famed because is related with an unsolver problem (see here in the page 345 this Notices of The AMS, The Riemann Hypothesis, by Conrey).
I would like to explore more easy facts of this space and their elements.
Question 1. What 's a good basis to explore this linear subspace of $L^2(0,1)$?
I say that since previous subset is defined as a subspace spanned, I would like to know a good basis to try explore questions about this subspace and their elements from the viewpoint of the real analysis and functional analysis (my main purpose is refresh issues of this theories). I thought in the trygonometric system or the Haar system for the unit interval, but notice that in this article from the Wikipedia's Page, the unit interval is the closed set $ \left[ 0,1 \right] $ (do a comparison with the open set $\left( 0,1 \right) $) of $L^2(0,1)$. I don't understand well previous cited unsolved problem to know if the fact to declare a such basis implies the thesis, (this is the veracity of such identity between spaces showed in page 345 of previous article) in any case I need a basis to do computations.
A main problem is that $\alpha$ is in an uncountable set, and I don't know what can I do with this subspace to put a basis in it. Or if the right is put a basis in the underlying Hilbert space $L^2(0,1)$.
My next question is if you want to provide to me a easy example of how do you compute the coordinates of the more easy function in this span endowed with the basis that you've choosen. In this Mathematics Stack Exchange you've the answer of another of my previous of questions, that provide us easy examples of such functions.
Question 2. Compute the coordinates of $\eta_{\frac{1}{2}}(t)$ in the basis that you've choosen in previous question. Thanks in advance.
I want to refresh these notions, and improve my abilities in real and functional analysis in the next season.