Whose basis is {1,sin(x),cos(x),sin(2x),cos(2x),…}? Whenever $f(x)$ is a (Riemann) integrable function on $[-\pi,\pi]$ we can define its Fourier series $f=a_0/2+\sum a_nsin(nx)+b_ncos(nx)$.But we give arbitrary sequence {$a_n$} and {$b_n$},I think $a_0/2+\sum a_nsin(nx)+b_ncos(nx)$ may not be integrable.Then let $R$={$f(x)|f(x)$ is Riemann integrable },$S$=span{1,sin(x),cos(x),sin(2x),cos(2x),…},one have $R\subset$S.We know {1,sin(x),cos(x),sin(2x),cos(2x),…} is a basis,but which space it is the basis for?
 A: You can use this basis as a basis for various spaces, not just $L^2$.
A general way to build a complete normed function space is to: 


*

*Think up a norm, like $\sup |f|$, or $\sup |f|+\sup |f'|$, or $\int |f|$, or $\sqrt{\int |f|^2 }$, or maybe $|f(0)|+\sqrt{\int e^{-x^2}|f'(x)|^2}$... 

*Pick a bunch of functions $f_n$ with finite norm. 

*Take the (algebraic) linear span of those functions, i.e., the set of all finite linear combinations. The elements of this set still have finite norm. The norm defines a metric on the set.

*Take the completion of the set obtained in 3. 

*You have a function space, in which the functions picked at step 2 form a spanning set. 

*If you want them to be a basis, understood as Schauder basis, there is one more thing to  check: you must be able to approximate any element of the space by  partial sums of a series $\sum c_n f_n$, not just by some sequence of linear combinations. The series should be unique, too. 


You can go through steps 1-5 using trigonometric functions in 2 and pretty much any norm imaginable. At step 6 some norms will be disqualified. For example, you get a basis in $L^p$ for $1<p<\infty$, but not for $L^1$ (see the Wikipedia article).
