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This question might be better asked on the Math.SE site but I feel it could be well placed here as well.

My textbook (Sturm-Liouville Theory and its Applications , Al Gwaiz) defines the vector space $C([a,b])$ as the space of complex continuous functions defined on the real interval $[a,b]$. It then states that this space is not a suitable inner product space for the study of Sturm-Liouville Theory because "it is not closed under limit operations".

The text then goes on to illustrate the difference between pointwise convergence and uniform convergence and this discussion culminates in the proof that uniform convergence of a sequence of continuous functions preserves continuity. That is, the uniform limit of continuous functions is continuous. We also learn that point-wise convergence does not preserve continuity however. This seems to indicate that the space $C([a,b])$ of continuous functions is in fact "closed under limit operations" if those limit operations are defined using uniform convergence as opposed to point-wise convergence. Surely this contradicts the authors claim that $C([a,b])$ "is not closed under limit operations"?

Thus my question is: How is using $L^2(a,b)$ beneficial over using $C([a,b])$ as our inner-product space for sturm-Liouville theory (or perhaps for all of physics more broadly) when both seems to be perfectly adequate spaces considering they are both closed under limit operations provided we define limit operations on $C([a,b])$ using uniform convergence?

Qmechanic
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SalahTheGoat
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    Related: https://physics.stackexchange.com/q/20822/2451 , https://physics.stackexchange.com/q/41719/2451 and links therein. – Qmechanic Sep 09 '21 at 08:38
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    Sturm-Liouville theory makes use of the inner product $\langle f,g\rangle:=\int_a^b f(x)g^\ast(x)\mathrm dx$. So it makes sense to use the type of convergence induced by this inner product. That is, $L^2$-convergence, in which $f_n\to f$ if $\Vert f-f_n\Vert_2\to0$, where $\Vert f\Vert_2:=\sqrt{\langle f,f\rangle}$. The continuous functions are not closed under this limit operation. – Vercassivelaunos Sep 09 '21 at 09:36

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