About completeness of the Fourier series. The Fourier series of a function is given by $$ \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos n \theta +  \sum_{n=1}^\infty b_n \sin n \theta . $$ Here what does the statement " $\sum_{n=1}^\infty b_n \sin n \theta $  is complete"  mean? And would you tell me how can I prove this statement?  
 A: I don't think it makes sense to call a Fourier series complete. 
But what you can call complete is the basis of your space. 
For example, if the functions you're computing Fourier series of are in $L^2(G)$, where $G$ is a compact Abelian topological group, one can say that the characters $\chi$ of $G$ are complete and it means two things: 
(i) that for every function $f$ in $L^2$ there exists a sequence of continuous characters $\chi_n \in \mathrm{Hom}(G, S^1)$ such that $\|f - \sum_{k=0}^n a_k \chi_k \|_{L^2} \xrightarrow{n \to \infty} 0$ and
(ii) that $\chi_n$ are orthonormal, that is, $\langle \chi_n , \chi_m \rangle = \delta_{nm}$ where $\langle \cdot, \cdot \rangle$ is the inner product of $L^2$, namely, $\langle f, g \rangle = \int_G f \;\overline{g}\; d \mu$ (where $\overline{g}$ denotes complex conjugation)
A: I doubt this statement makes sense. 
What does make sense is the statement that the trigonometric polynomials ($\sin(kx), \cos(kx), k\in \mathbb{N}$ and the constant function 1) form a complete orthogonal system (orthonormal with correct normalization) of functions in $L^2([0,2\pi])$, say. 
This then refers to the fact that each function in $L^2([0,2\pi])$ admits a unique representation like the one you wrote down with convergence w.r.t to the norm of $L^2$, i.e. 
$$ ||f(x) - \frac{a_0}{2} - \sum_1^n (a_n \cos (nx)+b_n \sin(nx))||_{L^2} \rightarrow 0 \,\, (n\rightarrow \infty) $$
A proof of this fact can be found in several standard texts on analysis, like e.g. Rudin's Principles of mathematical analysis.
A: Supposing you omit the $n=5$ term from the cosine series so $a_5\cos(5\theta)$ just isn't included.  Why would the set of functions you're adding up be incomplete?
The point is that there would be some functions that cannot be approached by series that omit that term.  One such function is $\cos(5\theta)$ itself.
"Approached" means convergence in the $L^2$ sense, i.e. $f_n\to f$ in $L^2$ means that $\displaystyle\int_0^{2\pi} |f_n(\theta) - f(\theta)|^2\,d\theta \to 0$ as $n\to\infty$.
And "some functions" means some $L^2$ functions, i.e. functions $f$ such that $\displaystyle\int_0^{2\pi} |f(\theta)|^2\,d\theta<\infty$.
