Lᵖ Spaces

The Lebesgue integral does something the Riemann integral never could: it measures the size of a function itself. If \int |f|\,d\mu is a finite number, that number is a legitimate "length" for f — and once functions have lengths, they have distances between them, and for one magic exponent even angles. Functions stop being graphs on a page and become vectors in a space, one with genuine, if infinite-dimensional, geometry.

That is the single idea of this page: the Lᵖ spaces. They are the vector spaces of integrable functions, made into normed spaces by the integral. They are complete, so limits behave. And the case p = 2 is a Hilbert space — the infinite-dimensional twin of Euclidean space, complete with orthogonality and projections. This is not an abstraction for its own sake: Lᵖ spaces are the native habitat of Fourier analysis, of partial differential equations, and of quantum mechanics, where a particle's state is a unit vector in L^2.

The Lᵖ norm and the space it builds

Fix a measure space (X, \mathcal{M}, \mu) and a real exponent 1 \le p < \infty. For a measurable function f define its Lᵖ norm as

\|f\|_p \,=\, \left( \int_X |f|^p \, d\mu \right)^{1/p}.

The p-th power inside, the integral, and the p-th root outside are exactly the ingredients that will make this behave like a length (the root is what earns the triangle inequality below). The space itself is everything with a finite such length:

L^p(\mu) \,=\, \bigl\{\, f \text{ measurable} : \|f\|_p < \infty \,\bigr\}.

At the other end sits p = \infty. Here the "length" is not an integral but a ceiling — the smallest bound that holds almost everywhere, the essential supremum:

\|f\|_\infty \,=\, \operatorname*{ess\,sup}_{x} |f(x)| \,=\, \inf\bigl\{\, M \ge 0 : |f| \le M \text{ a.e.} \,\bigr\}, \qquad L^\infty(\mu) = \{ f : \|f\|_\infty < \infty \}.

The word essential matters: \|f\|_\infty ignores what f does on any set of measure zero, so a function that is 10^{9} at a single point but bounded by 3 elsewhere has \|f\|_\infty = 3.

\|\cdot\|_p is almost a norm — but it fails one axiom on the nose. Recall the vanishing property of the integral:

This is a subtlety worth internalising early: a point of L^p is a whole family of functions agreeing a.e., glued together. We keep writing "f \in L^p" and calling it a function out of habit, but the object that actually lives in the space is its class.

The two inequalities that make it a space

Two theorems do all the heavy lifting. The first controls products, the second controls sums — and it is the second that certifies \|\cdot\|_p as a norm.

Let p, q \in (1, \infty) be conjugate exponents, meaning

\frac{1}{p} + \frac{1}{q} = 1.

Then for measurable f, g,

\|fg\|_1 = \int |fg|\,d\mu \;\le\; \|f\|_p\,\|g\|_q.

For 1 \le p \le \infty and f, g \in L^p,

\|f + g\|_p \;\le\; \|f\|_p + \|g\|_p.

This is the triangle inequality for \|\cdot\|_p. With it, the three norm axioms all hold — positivity (on a.e.-classes), \|cf\|_p = |c|\,\|f\|_p, and the triangle inequality — so (L^p, \|\cdot\|_p) is a genuine normed vector space, and d(f, g) = \|f - g\|_p is a genuine metric.

Minkowski for p > 1 is proved using Hölder: split |f+g|^p \le |f|\,|f+g|^{p-1} + |g|\,|f+g|^{p-1}, integrate, and apply Hölder to each term with exponent q = p/(p-1) on the trailing factor. The two theorems are a matched pair.

What does the "unit ball" look like?

A norm is fully pictured by its unit ball — the set of vectors of length \le 1. In the two-dimensional model space \mathbb{R}^2 with the p-norm \|(x,y)\|_p = (|x|^p + |y|^p)^{1/p}, the unit ball is \{|x|^p + |y|^p \le 1\}, and it changes shape dramatically with p. Drag the slider and watch:

At p = 1 the ball is a diamond (the taxicab metric); at p = 2 it is a perfectly round circle; as p \to \infty it swells into a square (the max-norm). The roundness at p = 2 is not cosmetic — it is the visible signature of the inner product, the one exponent where the norm comes from a dot product and rotations are symmetries. The same story plays out, invisibly, in the infinite-dimensional function spaces L^p: only L^2 has a "round" geometry with angles.

Completeness: Riesz–Fischer, and why the Lebesgue integral pays off

A normed space is only as useful as its limits. The decisive fact about L^p is that it is complete: every Cauchy sequence converges to a limit that is still in the space. A complete normed space is a Banach space.

For every 1 \le p \le \infty, the space L^p(\mu) is complete. That is:

This is exactly where the long climb through measure theory earns its keep. The naive alternative — the space of Riemann-integrable functions under the same \|\cdot\|_2 — is not complete: one can build a Cauchy sequence of nice Riemann-integrable functions whose limit is something like \mathbf{1}_{\mathbb{Q}}, which escapes the space entirely. Lebesgue's integral is precisely the completion that plugs those holes. Completeness is not a technical footnote; it is why L^2 works — why Fourier series converge to genuine functions, why the fixed-point and projection theorems of analysis apply, why the whole edifice stands.

L² is special: the one with angles

Among all the L^p, the case p = 2 is set apart, because its norm is born from an inner product:

\langle f, g \rangle \,=\, \int_X f \, \overline{g} \, d\mu, \qquad\text{so that}\qquad \|f\|_2 = \sqrt{\langle f, f \rangle} = \left(\int |f|^2\,d\mu\right)^{1/2}.

(The complex conjugate \overline{g} is there so \langle f, f\rangle = \int |f|^2 \ge 0 for complex-valued functions; for real functions it is just \int fg.) An inner product is far more than a norm — it encodes angles. Two functions are orthogonal when \langle f, g\rangle = 0, and with orthogonality come the full toolkit of Euclidean geometry in infinitely many dimensions: the Pythagorean theorem, orthogonal projection onto a subspace, and orthonormal bases.

In short: L^2 is a complete inner-product space, i.e. a Hilbert space — the infinite-dimensional analogue of Euclidean \mathbb{R}^n. The Cauchy–Schwarz inequality above is just Hölder at p = q = 2, and it is what lets us define \cos\theta = \tfrac{\langle f, g\rangle}{\|f\|_2\|g\|_2}. For p \ne 2 there is no such inner product and no notion of angle at all — a L^p for p \ne 2 is a Banach space but never a Hilbert space.

Worked examples: who is in, who is out

Example 1 — a power near the origin: f(x) = x^{-a} on (0,1)

Take a > 0 and ask when f(x) = x^{-a} lies in L^p(0,1). Compute the defining integral directly:

\int_0^1 |x^{-a}|^p \, dx = \int_0^1 x^{-ap}\, dx, \qquad\text{which is finite} \iff ap < 1 \iff a < \frac1p.

The singularity at 0 is only integrable if the exponent ap stays below 1. So x^{-1/2} \in L^1(0,1) but x^{-1/2} \notin L^2(0,1) (there ap = 1, borderline divergent). Larger p punishes the spike more.

Example 2 — the same power far out: f(x) = x^{-a} on (1,\infty)

Now the trouble is at infinity, and the condition flips:

\int_1^\infty x^{-ap}\, dx \;\text{ is finite} \iff ap > 1 \iff a > \frac1p.

A fat tail needs ap above 1 to decay fast enough. The two examples together are the classic warning: on the infinite line \mathbb{R} a single power can be integrable at neither end, and no inclusion L^p \subseteq L^q holds in general.

Example 3 — nesting on a finite measure space

When the whole space has finite measure — say [0,1] with \mu([0,1]) = 1 — the powers do nest, and the bigger exponent wins:

\mu(X) < \infty \ \text{ and } \ p_1 > p_2 \quad\Longrightarrow\quad L^{p_1}(\mu) \subseteq L^{p_2}(\mu).

(One line of Hölder: apply it to |f|^{p_2} \cdot 1 with exponent p_1/p_2, and the finiteness of \mu(X) supplies the needed factor.) So on [0,1], L^\infty \subset \cdots \subset L^2 \subset L^1 — every bounded function is square-integrable, every square-integrable function is integrable.

Example 4 — a norm you can compute

Let f be the two-step simple function on [0,1] equal to 3 on [0,\tfrac12) and 1 on [\tfrac12, 1]. Then

\|f\|_2 = \left(\int_0^1 |f|^2\,dx\right)^{1/2} = \left(3^2\cdot\tfrac12 + 1^2\cdot\tfrac12\right)^{1/2} = \sqrt{\tfrac{9}{2} + \tfrac12} = \sqrt{5}.

Its length as a vector in L^2[0,1] is \sqrt5 — a concrete number attached to a function, computed by exactly the same value-times-measure bookkeeping the Lebesgue integral gave us.

Yes — that is the whole secret. On [-\pi, \pi] the functions e_n(x) = \tfrac{1}{\sqrt{2\pi}}\,e^{inx}, for n \in \mathbb{Z}, are orthonormal in L^2: \langle e_n, e_m\rangle = \int_{-\pi}^{\pi} e_n \overline{e_m} = \delta_{nm}, a clean 1 when n = m and 0 otherwise. And they are complete: they span the whole space. So a function's Fourier series is nothing more exotic than its expansion in an orthonormal basis,

f = \sum_{n} \langle f, e_n\rangle \, e_n, \qquad \|f\|_2^2 = \sum_n |\langle f, e_n\rangle|^2,

with that second identity — Parseval's theorem — being the infinite-dimensional Pythagoras. The convergence \sum \to f is convergence in L^2, and it holds for every f \in L^2 precisely because the space is complete (Riesz–Fischer). The nineteenth century's mess over "does the Fourier series converge?" dissolves once you see it as a coordinate expansion in a Hilbert space.

Five traps at the door of Lᵖ: