Maybe I can give a less technical explanation!
When we study functions, we're interested in two types of properties they might have.
One is their regularity. This talks about how a function behaves locally, that is, in a small ball (or interval) around a particular point. The simplest regularity property is continuity, and we write the set of continuous functions from a space X to the real numbers R as C(X).
We could also care about differentiability, and we write the set of continuous functions with one continuous derivative as C^1(X), and similarly C^n(X) is the set of continuous functions with n continuous derivatives.
The best type of regularity we can hope for is that a function is smooth, that is, it has infinitely many continuous derivatives. Good examples: sin(x) and cos(x). We write C^infinity(X) for smooth functions on X.
These sets have a natural inclusion property: that is
C^infinity(X) is contained in C^n(X) for all n, is contained in C(X).
The common way of "measuring" regular functions is by the supremum norm I mentioned in the previous post.
The second type of property we care about is integrability. This is a global property, as it takes into account how a function behaves on the whole space, not just around in a small ball around each point.
The set of integrable functions (functions for which the integral of the absolute value is finite) is written L^1(X).
Some functions need to be squared before they are integrable and we thus write the set of square-integrable functions as L^2(X).
Similarly, L^p(X) is the set of functions whose integral is finite when we raise its absolute value to the pth power (their "p-integral" ).
We write L^infinity(X) for the set of functions which are bounded on X.
There is no such nice inclusion here.*
The common way of "measuring" p-integrable functions is by computing their p-integral and taking its pth root. On square-integrable functions we have the norm given by the square root of the integral of the square of the function and also an inner product. This inner product (as you wrote in your first post) is given by the integral of the product of two functions.
What about the relationship between regularity properties and integrability properties?
If the space X we are working in is infinite (like the real line R, for example), then there is not much relationship at all. Example: x^2 is smooth on R (all its derivatives exist and are continuous) but not integrable (its integral on the whole line is infinite). There are of course functions which are very regular and integrable; example: e^(-x^2) is smooth and integrable on R.
If the space X we're working on is compact (essentially, this means that X is bounded and contains its boundary, for example the closed interval [0,1]), then there is an inclusion: C(X) is included in all of the L^p(X) spaces because of the well-known result that a continuous function on a compact set is bounded.
Now the punchline: on an infinite space, there isn't much point measuring regular functions with their p-integrals as these are essentially trying to characterize functions whose behaviour is good locally by their global properties. On a compact space, both supremums and integrals make sense (because continuous functions are integrable on a compact space), but all the integrals tell us is about global things, where we are interested in local things.
Sorry it's so long, but maybe it helps?
*It's not immediately obvious why; I can explain more if you like?