**Theorem 1** The functions $x^ne^{-x^x}$, $n=0,1,2,\cdots,$ are complete in $L^2(\mathbb{R},dx)$. //Proof// Let $f\in L^2(\mathbb{R},dx)$ such that $$\int_{\mathbb{R}} f(x)x^n e^{-x^2} dx =0,\quad,n=0,1,2,\cdot.$$ It is easy to see that the integral $$F(z)=\int_{\mathbb{R}} f(x)e^{ixz-x^2}dx$$ is absolutely convergent for all $z\in\mathbb{C}$ and, therefore, defines an entire function. Direct computation shows that $$F^{(n)}(0) = i^n\int_{\mathbb{R}} f(x)x^n e^{-x^2} dx =0,\quad,n=0,1,2\cdot,$$ so that $F(z)=0$ for all $z\in\mathbb{C}$. We then can deduce that $f=0$. **Theorem 2** The set $\{e_n\| n\geq 1\}$ defined by $$e_n(x)=\sqrt{\frac{2}{\pi}}sin(nx),$$ and the set $\{f_n\| n\geq 0\}$ defined by $$f_0=\sqrt{\frac{1}{\pi}},\quad f_n(x)\sqrt{\frac{2}{\pi}}cos(nx),\quad\forall\,n\geq 1,$$ are both orthonormal basis of $L^2([0,\pi])$. === Reference [Tak08] L. A. Takhtajan, Quantum Mechanics for Mathematicians, Graduate Studies in Mathematics, Vol. 95, American Mathematical Society, 2008