Fourier Transform Motivation/Derivation

Question

this is my first ever Math SE question, and I am wondering how one can go about rigorously explaining the Fourier Transform. I believe it is connected to Fourier Series, but I can't comprehend the connection. To find the Fourier Series (complex version) of a function $f(x)$ with period $2T, T>0$, one derives an orthonormal basis for $L^2[-T,T]$ with inner product $$\langle u(x),v(x) \rangle=\int_{0}^{1}u(x)\overline{v(x)}dx.$$ The basis turns out to be $$B=\left(\frac{1}{\sqrt{2T}},\frac{e^{\frac{inx\pi}{T}}}{\sqrt{2T}},\frac{e^{\frac{-inx\pi}{T}}}{\sqrt{2T}}: n\in \mathbb{N}\right),$$ which I understand. The Fourier series is the span of these basis elements, and the constants are Fourier coefficients computed by three different inner products. I read about the Fourier transform, which is to supposed deal with the case $T \rightarrow {\infty}.$ My question is can we motivate the Fourier Transform in a similar way to the Fourier series, which is to come up with an orthonormal basis (for $L^2(\mathbb{R})$ I believe) and write the transform as the span of such basis elements ? If so, since the transform is described by an integral, is the span essentially a Riemann sum in this case? I know there are other questions like this, but I just am struggling to understand the motivation of the Fourier Transform.