First of all, what is a square-integrable function? Wikipedia provides this definition:
In mathematics, a square-integrable function, also called a quadratically integrable function, is a real– or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, if
then ƒ is square integrable on the real line . One may also speak of quadratic integrability over bounded intervals such as [0, 1].
The important part is seeing that if the integral of the square of a [Real-valued] function is finite, then the absolute value of the function must converge to zero at +/- infinity, by common sense.
Now the thing that really got me thinking about this was, there have been a lot of good things that have run their courses in my lifetime thus far. And I got thinking about this, because some of these were really good things that I really cherished. In the end, the good things were still crushed by time. Of course though, so are the bad things. If our independent variable was time, than any relevant function of time — any good or bad thing that was time-dependent — was a square-integrable function. There was a beginning and there was an end. At some point, the function hit zero, and it didn’t come back.
Here is the kicker though. Any function that is constant cannot be square-integrable. Just use common sense. So this means that if anything looks time-independent, we are just Taylor expanding and neglecting higher-order terms. So really, everything is time-dependent, if we are to believe that everything is square-integrable.
Consider my prior post about friendships and chemical bonding. In the shortest timescales — to the lowest order I suppose — this is accurate. Now I’m going to claim that if we add a higher-order term, we start involving time dependence. I don’t know the nature of this time-dependence in chemistry (I’m guessing there is one because it is dynamic equilibrium after all), but can friendships decay over time? I mean, if we are to truly believe the idea of square-integrablility, then it has to be. And then our job, I guess as social beings, is not only to maximize our kinetic and thermodynamic affinities towards social interaction — it includes minimizing some characteristic decay parameter that states how fast the friendship “goes to zero”, so to speak.
Considering how much analysis and thought I’ve given to the thermodynamic and kinetic viewpoints, it’s easy to see how I can overlook time dependencies. But it is of the utmost importance; it is insufficient to make friends if you cannot keep them. This is one of the lessons I learned in China. My mom not only has a lot of friends, she has a lot of friends who would still sacrifice time and effort for her even after 20 years of not seeing each other. Evidently, she not only minimized any temporal friendship fluctuations, she minimized spatial fluctuations. Now that’s pretty impressive. The problem is, it’s hard for me. I have trouble keeping friendships intact over space and time. This semester is reminding me of my shortcomings. It’s hard.
There’s a lot more we can learn from square-integrable functions. For one, we have to appreciate our fortunes, and not dwell on our misfortunes — both disappear given time. But at least for me, square-integrable functions tell me that I have a lot to learn and practice.