Prove that the systems are orthonormal bases in the space $L_2[0, \pi]$. The question is as follows: Prove that $\{\sqrt{\frac{2}{π}}\sin({(n-\frac{1}{2}) t})\}_{n =1, ...,\infty}$ and $\{\sqrt{\frac{2}{π}}\cos({(n-\frac{1}{2}) t})\}_{n =1, ...,\infty}$ are orthonormal bases in $L_2[0,π]$.
I have shown that the systems are orthonormal. It remains to show that this is the basis.
I thought to show completeness by the fact that there is no function other than zero and orthogonal to all the functions of this system. But something doesn't add up.
Can someone shed some light on what I am doing wrong, or thinking for that matter?
 A: Consider a unitary isomorphism $Uf(x)=e^{-\frac{ix}{2}}f(x)$ on the $L_2[-\pi,\pi]$. Applying it to the standard basis of $L_2[-\pi,\pi]$, i.e. to $e^{ikx}$, $k\in\mathbb{Z}$, we obtain the system $e^{i(k-\frac{1}{2})x}$, $k\in\mathbb{Z}$ that will also be a basis of $L_2[-\pi,\pi]$.
Now, consider $f(x)\in L_2[0,\pi]$ and denote $F(x)$ -- odd extension of a function $f$ to a $L_2[-\pi,\pi]$. Let $\displaystyle\int\limits_{0}^\pi f(x)\sin\left(k-\frac{1}{2}\right)xdx=0$ for all $k\in\mathbb{N}$. Then $$0=\int\limits_{0}^\pi f(x)\sin\left(k-\frac{1}{2}\right)xdx=\frac{1}{2}\int\limits_{-\pi}^\pi F(x)\sin\left(k-\frac{1}{2}\right)xdx=$$$$=\frac{1}{2}\int\limits_{-\pi}^\pi F(x)\sin\left(k-\frac{1}{2}\right)xdx-\frac{i}{2}\int\limits_{-\pi}^\pi F(x)\cos\left(k-\frac{1}{2}\right)xdx=-\frac{i}{2}\int\limits_{-\pi}^\pi F(x)e^{i(k-\frac{1}{2})x}dx.$$
Thus, $F$ belongs to the orthogonal complement to the linear span of $e^{i(k-\frac{1}{2})x}, \;k\in\mathbb{N}$, i.e. $F$ belongs to the closure of the linear span of $e^{i(k-\frac{1}{2})x},\;k=0,-1,-2,...$ If $F(x)=\sum c_ke^{i(k-\frac{1}{2})x}$, then, due to oddness, $\sum c_k\cos\left(k-\frac{1}{2}\right)x=0$. By multiplying this equality scalarly on $\cos\left(k-\frac{1}{2}\right)x$ on the $[0,\pi]$, we get that all $c_k=0$, thus $F(x)\equiv0$.
Another way to solve the problem is to choose a suitable Sturm-Liouville problem, the solution of which is the required systems of functions.
