Mathematics.

function spaces

Lᵖ Spaces

Measure Theory85 minDifficulty8 out of 10

You should know: lebesgue integral, measurable functions, measure spaces

Overview

The L^p spaces (1 <= p <= ∞) are Banach spaces of measurable functions on a measure space (X, F, mu), equipped with the L^p norm. They provide the natural function spaces for Fourier analysis, PDEs, probability theory, and functional analysis. The cases p = 1 (integrable functions), p = 2 (square-integrable, a Hilbert space), and p = ∞ (essentially bounded) are most frequently encountered.

Intuition

L^p spaces measure 'how large' functions are in an average sense controlled by p. Large p punishes large peaks more severely. L^1 measures total area; L^2 measures total squared amplitude (energy, in physics); L^∞ measures the essential supremum. The triangle inequality (Minkowski's inequality) makes the L^p norm well-behaved, and completeness (every Cauchy sequence converges) makes L^p a Banach space.

Formal Definition

Definition

Let (X, F, mu) be a measure space and 1 <= p < ∞.

∥f∥p=(∫X∣f∣p dμ)1/p\|f\|_p = \left(\int_X |f|^p\, d\mu\right)^{1/p}
L^p norm (1 <= p < ∞)
Lp(μ)={f:X→R measurable:∥f∥p<∞}/∼L^p(\mu) = \left\{f : X \to \mathbb{R} \text{ measurable} : \|f\|_p < \infty\right\} \big/ {\sim}
L^p space (functions identified a.e.)
∥f∥∞=ess sup⁡x∈X∣f(x)∣=inf⁡{M≥0:∣f(x)∣≤M a.e.}\|f\|_\infty = \operatorname{ess\,sup}_{x \in X} |f(x)| = \inf\{M \ge 0 : |f(x)| \le M \text{ a.e.}\}
L^∞ (essential supremum) norm
1p+1q=1(p,q are Ho¨lder conjugates)\frac{1}{p} + \frac{1}{q} = 1 \quad (p,q \text{ are H\"older conjugates})
Holder conjugate exponents
∫X∣fg∣ dμ≤∥f∥p∥g∥q(Ho¨lder’s inequality)\int_X |fg|\, d\mu \le \|f\|_p \|g\|_q \quad \text{(H\"older's inequality)}
Holder's inequality
∥f+g∥p≤∥f∥p+∥g∥p(Minkowski’s inequality)\|f + g\|_p \le \|f\|_p + \|g\|_p \quad \text{(Minkowski's inequality)}
Minkowski's inequality — triangle inequality for L^p

Notation

NotationMeaning
∥f∥p\|f\|_pL^p norm of f
Lp(μ)L^p(\mu)L^p space with measure mu
Lp(a,b)L^p(a,b)L^p space on (a,b) with Lebesgue measure
ess sup⁡\operatorname{ess\,sup}Essential supremum — supremum ignoring null sets
(Lp)∗≅Lq(L^p)^* \cong L^qDual space of L^p is L^q (1 < p < ∞)

Properties

L^p is a Banach space

(Lp(μ),∥⋅∥p) is complete for every 1≤p≤∞(L^p(\mu), \|\cdot\|_p) \text{ is complete for every } 1 \le p \le \infty

Condition: Completeness proven via Riesz–Fischer theorem

L^2 is a Hilbert space

⟨f,g⟩=∫Xfg dμ makes L2(μ) a Hilbert space\langle f, g \rangle = \int_X fg\, d\mu \text{ makes } L^2(\mu) \text{ a Hilbert space}

Containment for finite measure spaces

μ(X)<∞,;p≤q  ⟹  Lq(μ)⊆Lp(μ)\mu(X) < \infty,; p \le q \implies L^q(\mu) \subseteq L^p(\mu)

Dual space

(Lp(μ))∗≅Lq(μ) for 1≤p<∞,;1/p+1/q=1(L^p(\mu))^* \cong L^q(\mu) \text{ for } 1 \le p < \infty,; 1/p + 1/q = 1

Condition: Requires sigma-finiteness for p = 1

Dense subsets

Simple functions are dense in Lp(μ) for 1≤p<∞\text{Simple functions are dense in } L^p(\mu) \text{ for } 1 \le p < \infty

Worked Examples

  1. Compute the L^1 norm.

    ∥f∥1=∫01x−1/2 dx=[2x1/2]01=2<∞\|f\|_1 = \int_0^1 x^{-1/2}\, dx = \left[2x^{1/2}\right]_0^1 = 2 < \infty

  2. Compute the L^2 norm squared.

    ∥f∥22=∫01x−1 dx=[ln⁡x]01=∞\|f\|_2^2 = \int_0^1 x^{-1}\, dx = \left[\ln x\right]_0^1 = \infty

  3. So f ∈ L^1(0,1) but f ∉ L^2(0,1), showing L^2 ⊄ L^1 in general (though on finite measure spaces L^2 ⊆ L^1).

Answer: f(x) = x^{-1/2} is in L^1(0,1) but not in L^2(0,1).

Practice Problems

Difficulty 7/10

For which values of alpha is f(x) = |x|^alpha in L^p(-1,1)?

Difficulty 8/10

Prove Holder's inequality for p = 2 (Cauchy–Schwarz inequality): ∫|fg| dmu <= ||f||_2 ||g||_2.

Difficulty 8/10

Give an example showing that L^2(R) ⊄ L^1(R).

Common Mistakes

Common Mistake

L^p ⊆ L^q whenever p < q, on any measure space

On finite measure spaces L^q ⊆ L^p for p <= q. On infinite measure spaces (like R with Lebesgue measure), neither L^p nor L^q contains the other in general.

Common Mistake

The L^p norm is defined on individual functions

Strictly, L^p consists of equivalence classes of functions where f ~ g if f = g a.e. The 'norm' ||f||_p is zero for all a.e.-zero functions, so we must identify them to get a genuine norm.

Common Mistake

Pointwise convergence implies L^p convergence

Pointwise (or a.e.) convergence does not imply L^p convergence. DCT gives L^p convergence under additional domination. The example f_n = n · 1_{(0,1/n)} converges a.e. to 0 but ||f_n||_1 = 1.

Quiz

L^2(mu) is distinguished from other L^p spaces because:
The Holder conjugate of p = 4 is:
On a finite measure space, if p < q, then:

Historical Background

The L^p spaces emerged naturally from Lebesgue's integration theory in the early 20th century. Frigyes Riesz systematically studied L^p in 1910, proving what is now called the Riesz–Fischer theorem (L^2 is complete). The H"older and Minkowski inequalities, essential to establishing the norm structure, were classical results adapted to the Lebesgue setting.

  1. 1910

    Riesz proves completeness of L^2 (Riesz-Fischer theorem)

    Frigyes Riesz, Ernst Fischer

  2. 1913

    Riesz extends to general L^p spaces

    Frigyes Riesz

  3. 1927

    Riesz representation for L^p dualities clarified

    Frigyes Riesz

Summary

  • L^p(mu) is the Banach space of (equivalence classes of) measurable functions with finite L^p norm ||f||_p = (∫|f|^p dmu)^{1/p}.
  • L^2 is the only L^p space that is a Hilbert space, with inner product ⟨f,g⟩ = ∫fg dmu.
  • Holder's inequality: ∫|fg| dmu <= ||f||_p ||g||_q where 1/p + 1/q = 1.
  • Minkowski's inequality: ||f+g||_p <= ||f||_p + ||g||_p (triangle inequality for L^p).
  • On finite measure spaces L^q ⊆ L^p for p <= q; this fails on infinite measure spaces.

References

  1. BookFolland — Real Analysis, 2nd ed. (1999), §6.1–§6.2
  2. BookRudin — Real and Complex Analysis, 3rd ed. (1987), Chapter 3
  3. WebsiteWikipedia — Lp space