Plane wave basis in a box. Inconsistent terminology about Fourier series and transform

I will refer to the one-dimensional equivalent of the problem in the picture for simplicity. At the end of the picture, the author argues that the Fourier transform of the function $psi(x)$ is the expression for the coefficients $a_k$ of the Fourier series. Is this a valid and general statement?

The notes seem to be mixing the concepts of Fourier series and Fourier transform. $psi$ is argued to be writable as a Fourier series, does it mean that $psi$ is periodic? If it were, the coefficients expresion of the Fourier expansion of a periodic function $f$ with period $L$ should follow a different formula:
$$ f(y) =sum_{k} c_k e^{iky}$$
$$c_k =frac{1}{L}int_{-L/2}^{L/2} dy f(y) e^{-iky}tag{1}$$

from that of the Fourier transform of a non-periodic function $g$:

$$hat{g}=h_1int_{-infty}^{infty}dy g(y) e^{-iky} tag{2}$$
(where $h_1$ is some constant according to the adopted convention, such that the respective constant $h_2$ in the expresion for the inverse formula should verify $h_1h_2=frac{1}{2pi}$) which clearly doesn’t match with (1)

So translating the author’s statement to 1 dimension and pluging the expresion of $phi_k(x)$:In the problem V is a cubic box $[-L/2,L/2]^3$, so in 1 dimension it is the segment $[-L/2,L/2]$,
$$ psi(x)=sum_{k} a_k phi_k(x) =sum_{k} a_k e^{ikx}$$ and $$ a_k=(phi_k,psi) =int_{[-L/2,L/2]} dy phi_k^*(y)psi(y)= frac{1}{sqrt{L}}int_{-L/2}^{L/2}dy psi(y) e^{-iky} tag{1′}$$

From the stament of the author, I am confused about

a) If $a_k$ are the coefficients of a Fourier series, why don’t we have a $frac{1}{L}$ factor in (1′) , just like in (1) instead of $frac{1}{sqrt{L}}$ ?
Besides I don think $psi$ is periodic, even though the formula looks pretty much alike

b) If at the same time $a_k$ is the Fourier transform of $psi$,from the definition of Fourier transform , I should have something like
$$a_k=hat{psi}=h_1int_{-infty}^{infty}dy psi(y) e^{-iky}tag{1”}$$, which also fails to agree with (1) or (1′) because of the limits of integration and because of the constant h_1 which is normally taken to be $h_1=1$( implying $h_2=1/2pi$) an not $frac{1}{sqrt{L}}$

Can someone clear this up?

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