Generalized Fourier Coefficients

Assume we have an ON basis ${| e_{i} ⟩}$ for $L_{2}^{w} (a, b)$ . We can write, $| f ⟩ = \sum_{i} f^{i} | e_{i} ⟩ = \sum_{i} | e_{i} ⟩ ⟨ e^{i} | f ⟩$ . Then, $f (x) = ⟨ x | f ⟩ = \sum_{i} ⟨ x | e_{i} ⟩ ⟨ e^{i} | f ⟩ = \sum_{i} f^{i} e_{i} (x)$ . $f^{i} = \int_{a}^{b} d x w (x) e_{i}^{*} (x) f (x)$ .

$f^{i}$ are called fourier coefficients or generalized fourier coefficients.

Thus, $ℓ_{2}$ and $L_{2}^{w} (a, b)$ are isomorphics, i.e. the same.

Hence, given an orthonormal basis, a vector in $L_{2}^{w}$ is characterized by an infinite sequence $(f^{1}, f^{2}, \dots)$ , i.e. a vector in $ℓ_{2}$ , for which their sum of swaures converges to a finite value.

Therefore, all Hilbert spaces are isomorphic to $ℓ_{2}$ .

(A bit handwavy since we have infinite dimensional vector spaces).

Identity from Generalized Fourier Coefficients

$I = \sum_{i} | e_{i} ⟩ ⟨ e_{i} |$ .

$⟨ x | I | x^{'} ⟩ = \sum_{i} e_{i}^{*} (x) e_{i} (x^{'}) = δ (x - x^{'}) / w (x)$ .