Lp space
Imagine you want to group functions not by their formula but by how much total ‘energy’ they carry. Squaring a function and adding up the area underneath measures one kind of energy; taking the p-th power measures a whole family of them. The L^p space is the collection of all functions for which this total — the integral of |f|^p — comes out finite. Functions whose p-th power has finite integral are ‘tame’ in a way that ordinary boundedness does not capture: a function can blow up to infinity at a point and still belong to L^p, as long as the spike is thin enough.
Fix a measure space (X, M, mu) and a real number p with 1 <= p < infinity. The space L^p(X, mu) consists of all measurable functions f for which the integral of |f|^p over X is finite. To make the norm behave correctly we identify any two functions that agree almost everywhere — that is, off a set of measure zero — so an element of L^p is strictly speaking an equivalence class of functions, not a single function. Equipped with the L^p norm, it becomes a normed vector space, and in fact a complete one (a Banach space).
Two warnings. First, L^p membership depends on the whole space and the measure, not just local behavior: 1/x is not in L^1(0, infinity) because of slow decay at infinity, yet 1/sqrt(x) is in L^1(0, 1) despite blowing up at 0. Second, the spaces are not nested in general; on an infinite-measure space a function can lie in L^2 but not L^1, while on a finite-measure space the inclusions run the other way (larger p gives a smaller space).
On (0, 1) with Lebesgue measure, f(x) = x^(-1/3) satisfies the integral of |f|^p = the integral of x^(-p/3), which is finite exactly when p/3 < 1, i.e. p < 3. So f is in L^p(0,1) for every p < 3 but in no L^p with p >= 3.
Membership in L^p is a sharp threshold tied to how fast the function blows up.
L^2 is the only L^p that is a Hilbert space — its norm comes from an inner product, namely the integral of f times the conjugate of g. That extra geometric structure (angles, orthogonality, projections) is why L^2 dominates Fourier analysis and quantum mechanics.