In various applications where we are dealing with electromagnetic or acoustic waves, such as medical imaging, we use the space L 2 or the related Sobolev space W 1 , 2 . I have never seen a reason given for this. Does anyone know why these spaces are preferred over say L 1 spaces?

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In various applications where we are dealing with electromagnetic or acoustic waves, such as medical imaging, we use the space       L    2   or the related Sobolev space       W          1      ,      2      . I have never seen a reason given for this. Does anyone know why these spaces are preferred over say       L    1   spaces?

Haggar72 · Accepted Answer

In Quantum Mechanics the inner product between two functions       ϕ    1   and       ϕ    2   is given by  ⟨      ϕ    1    (  x  )      |        ϕ    2    (  x  )  ⟩  =  ∫  d  x        ϕ    1    ∗    (  x  )      ϕ    2    (  x  )    ,and this inner product has a very special place dedicated to it in the theory. Namely, the equation  ⟨      ϕ    1    (  x  )      |        ϕ    1    (  x  )  ⟩  =  ∫  d  x        |        ϕ    1    ∗    (  x  )            |        2    =  1must be true for any wavefunction that makes sense (i.e, wavefunctions must be normalizable). This shows already the importance of       L    2   spaces in QM, let aside its importante in Fourier analysis and harmonic theory. Mathematically,       L    2   spaces are actually the only       L    p   spaces that are also Hilbert spaces, where you can apply the extremely important Riesz Representation Theorem and say that to every ket there&#039;s a unique bra (modulo the usual equivalence class of       L    p   spaces). Albeit some non-normalizable wavefunctions (like position and momentum, which require rigged spaces) are also very important to the theory, most of QM is built upon the notion of       L    2   spaces.

In various applications where we are dealing with electromagnetic or acoustic waves, such as medical

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