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Question: Is there an intuitive way to think about mean square convergence of functions?

(mean square convergence = limit in quadratic mean = limit in $L^2$)

An equivalent concern is an intuitive way to think about functions which are "close" in $L^2$ to the constant zero function. (Since $X_n \to X$ in $L^2$ implies that $X_n - X \to 0$ in $L^2$.)


Motivation:
I want a simple way to explain to people that the mode of convergence used in the definitions of stochastic integrals is meaningful, ideally by appealing somehow to geometric or physical intuition.

In other words, I want to be able to think of or draw a picture which suggests a simple analogy relating convergence in mean square to something people might be more familiar with -- maybe waves or surfaces with elliptic curvature, or something like this.


Background: Both pointwise and uniform convergence of functions are easy to visualize as curves in the plane or space getting closer to each other.

However, seemingly unless we are talking about holomorphic functions, mean square convergence does not necessarily imply either of the above modes of convergence, as is often pointed in introductory discussions of Fourier series.

Is there some way to leverage one of the following notions to make mean square convergence seem like an intuitively appealing and meaningful concept, rather than a last-resort backup when we can't get pointwise or uniform convergence to work?

  • spectral decomposition
  • Karhunen-Loeve decomposition
  • variance, covariance
  • square integrable functions represent finite energy in quantum mechanics
  • orthogonality/orthogonal decompositions
  • weak stationarity
  • elliptic operators
  • quadratic curves and/or surfaces

and so on. This question is clearly related: Why consider square-integrable functions?

For example, what sort of everyday phenomenon should pictures like the following evoke? enter image description here
(From Wikipedia)

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    $\begingroup$ $f_n \to 0$ in $L^2$ if the area under $f_n$ is becoming small, where area far from the $x$-axis counts more than area near it. $\endgroup$ – Nate Eldredge Aug 17 '16 at 3:27
  • $\begingroup$ @NateEldredge Would this have something to do with the function "losing energy"? Right now I am thinking of elastic curves which minimize "bending energy" which is a quadratic term, see here: digitalcommons.georgiasouthern.edu/cgi/… Or maybe some other variational exercise resembling wave dissipation ("variation" in the sense of calculus of variations). $\endgroup$ – Chill2Macht Aug 17 '16 at 3:34
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A lot of analysts would think about $L^2$ convergence deliberately in terms of what it isnt, because you are often in a situation where you "only" have $L^2$ but might have or might want to have better, smoother convergence.

So typically I think, well it's not smooth or uniform or even defined pointwise everywhere

So I would say that a function that is mostly pointwise close to zero but whose graph has some tall, thin spikes is my typical picture. i.e. since it implies convergence in measure you have to be small on a large set but in the complement, on a small set, you may take very large values. In fact you might mentally replace the spikes with just regions where it is suddenly pointwise large.

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Real and complex $L^2$ spaces can be reduced to infinite-dimensional real and complex Euclidean spaces, where the coordinates in the Euclidean sequences $\{ x_{\alpha} \}_{\alpha\in\Lambda}$ are $\langle x,e_{\alpha}\rangle$ for an orthonormal basis $\{e_{\alpha} \}_{\alpha\in\Lambda}$. That is, $x_{\alpha}$ is the length of the projection of $x$ along the coordinate unit direction vector $e_{\alpha}$. The correspondence is isometric. So, if you understand convergence in $\ell^2$ Euclidean spaces, then you understand convergence in $L^2$ spaces, and vice versa.

A generic small element in an $L^2$ space is a scaled version of a generic large element.

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