Orthonormal basis in $L^2(\mathbb R)$ My Questions are:

(1)What are the ideas to prove  $ \{e^{2\pi i m \cdot}: m\in \mathbb Z\}$
forms an orthonormal
basis (ONB) for $L^2([0, 1))$?


(2) How to generalize the first part on $L^2(\mathbb R)$. Specifically,  can we expect that
$$\{\chi_{[0,1)}(\cdot-j)e^{2\pi i m \cdot}: j,m \in \mathbb Z\}$$
is an ONB for $L^2(\mathbb R)$

where
$\chi_{[0,1)}$ is the characteristic function of $[0,1)$ ?
 A: For (1), it suffices to show that a dense linear subspace $V$ of $L^2 [0,1)$ is contained in the closure of the linear subspace spanned by the functions $e^{2 i \pi m} : m \in \mathbb Z$.  You may take for $V$ the space of all smooth functions $\mathbb R \rightarrow \mathbb C$ which are $\mathbb Z$-periodic (that is, $f(x+n) = f(x)$ for all $x \in \mathbb R, n \in \mathbb Z$), restricted to $[0,1)$.  For such functions $f$, it follows from basic results in Fourier analysis that
$$f(x) = \sum\limits_{n \in \mathbb Z} c_n e^{2\pi i nx} \tag{1}$$
for all $x \in \mathbb R$, where $c_n = \int_0^1 f(x) e^{- 2\pi i nx}dx$.  The convergence on the right hand side is absolute, uniform on $\mathbb R$, and the right hand side also converges to the left in the Hilbert space $L^2[0,1)$ with its norm topology.
For general $f \in L^2[0,1)$, the formula (1) holds with uniqueness of expression for the coefficients $c_n$ (with the same formula), except it is no longer a pointwise limit for all $x \in [0,1)$ (indeed, elements of $L^2[0,1)$ are not really functions on $[0,1)$, but rather equivalence classes of functions, so it doesn't make sense to talk about pointwise convergence everywhere); the convergence only holds in the norm topology.  The coefficients $c_n$ satisfy $\sum\limits_{n \in \mathbb Z} |c_n|^2 < \infty$, and conversely, such a collection of complex numbers $c_n : n \in \mathbb Z$ uniquely determines an element of $L^2[0,1)$.
I don't know about (2).
