In the context of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where $Y_\ell^m( \theta , \varphi )$ are the Laplace spherical harmonics.

The context here is important because this equality holds only in the sense of the $L^2$-norm.

This expansion holds in the sense of mean-square convergence — convergence in [[Lp space|L²]] of the sphere — which is to say that

$$\lim_{N\to\infty} \int_0^{2\pi}\int_0^\pi \left|f(\theta,\varphi)-\sum_{\ell=0}^N \sum_{m=- \ell}^\ell f_\ell^m Y_\ell^m(\theta,\varphi)\right|^2\sin\theta\, d\theta \,d\varphi = 0.$$

So in general this limit is NOT pointwise? So I can't say that the value at a point of the function equals the value of the expansion at the same point?

If so, why it's usually stated out of the context of the structure of Hilbert space, that a bounded function or a square integrable function on the unit sphere can be expanded with Spherical Harmonics if it's not pointwise? I mean in some context outside the Hilbert space, where I am not interested in their square integral.

Furthermore, if it's not pointwise, but only in the norm, am I allowed to sum term by term two different functions, with two different expansions, like in the quantum scattering problem?

In the context of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where $Y_\ell^m( \theta , \varphi )$ are the Laplace spherical harmonics.

The context here is important because this equality holds only in the sense of the $L^2$-norm:

$$\lim_{N\to\infty} \int_0^{2\pi}\int_0^\pi \left|f(\theta,\varphi)-\sum_{\ell=0}^N \sum_{m=- \ell}^\ell f_\ell^m Y_\ell^m(\theta,\varphi)\right|^2\sin\theta\, d\theta \,d\varphi = 0.$$

Do we also have pointwise convergence almost everywhere?

In the contex of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where $Y_\ell^m( \theta , \varphi )$ are the Laplace spherical harmonics.

The context here is important because this equality holds only in the sense of the $L^2$-norm.

This expansion holds in the sense of mean-square convergence — convergence in [[Lp space|L²]] of the sphere — which is to say that

$$\lim_{N\to\infty} \int_0^{2\pi}\int_0^\pi \left|f(\theta,\varphi)-\sum_{\ell=0}^N \sum_{m=- \ell}^\ell f_\ell^m Y_\ell^m(\theta,\varphi)\right|^2\sin\theta\, d\theta \,d\varphi = 0.$$

So in general this limit is NOT pointwise? So I can't say that the value at a point of the function equals the value of the expansion at the same point?

If so, why it's usually stated out of the context of the structure of Hilbert space, that a bounded function or a square integrable function on the unit sphere can be expanded with Spherical Harmonics if it's not pointwise? I mean in some context outside the Hilbert space, where I am not interested in their square integral.

Furthermore, if it's not pointwise, but only in the norm, am I allowed to sum term by term two different functions, with two different expansions, like in the quantum scattering problem?

In the context of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where $Y_\ell^m( \theta , \varphi )$ are the Laplace spherical harmonics.

The context here is important because this equality holds only in the sense of the $L^2$-norm.

This expansion holds in the sense of mean-square convergence — convergence in [[Lp space|L²]] of the sphere — which is to say that

$$\lim_{N\to\infty} \int_0^{2\pi}\int_0^\pi \left|f(\theta,\varphi)-\sum_{\ell=0}^N \sum_{m=- \ell}^\ell f_\ell^m Y_\ell^m(\theta,\varphi)\right|^2\sin\theta\, d\theta \,d\varphi = 0.$$

So in general this limit is NOT pointwise? So I can't say that the value at a point of the function equals the value of the expansion at the same point?

If so, why it's usually stated out of the context of the structure of Hilbert space, that a bounded function or a square integrable function on the unit sphere can be expanded with Spherical Harmonics if it's not pointwise? I mean in some context outside the Hilbert space, where I am not interested in their square integral.

Furthermore, if it's not pointwise, but only in the norm, am I allowed to sum term by term two different functions, with two different expansions, like in the quantum scattering problem?