L^p Spaces — Mathematics Knowledge System

Overview

The L^p spaces (for 1 <= p <= infinity) are the fundamental function spaces of analysis. They consist of (equivalence classes of) measurable functions f whose p-th power of the absolute value is integrable: integral |f|^p < infinity. Equipped with the L^p norm, these spaces are Banach spaces (complete normed spaces). The duality (L^p)* = L^q (where 1/p + 1/q = 1) underlies Holder's inequality and is central to harmonic analysis and PDEs. L^2 is the unique Hilbert space among them.

Intuition

The L^p norm measures the 'size' of a function by integrating its p-th power. For p=1, it is the total variation (integral of |f|); for p=2, it is the energy (integral of f^2) — the natural inner product space. For p=infinity, it is the essential supremum. Larger p emphasises large peaks more. Holder's inequality generalises the Cauchy-Schwarz inequality and says: functions in L^p and L^q (conjugate exponents) can be paired via the L^1 inner product integral fg, with the product bounded by ||f||_p ||g||_q.

Formal Definition

Definition

Let (X, mu) be a measure space and 1 <= p < infinity. The space L^p(X, mu) consists of (equivalence classes of) measurable functions f: X -> R (or C) with ||f||_p = (integral |f|^p d mu)^{1/p} < infinity. For p = infinity, L^inf consists of essentially bounded functions with ||f||_inf = ess sup |f|. Two functions are equivalent if they agree almost everywhere. L^p(X, mu) is a Banach space for every 1 <= p <= infinity.

\|f\|_p = \left(\int_X |f|^p\,d\mu\right)^{1/p}, \quad 1 \le p < \infty

L^p norm

\|f\|_\infty = \mathrm{ess\,sup}\, |f| = \inf\{M : |f(x)| \le M \text{ a.e.}\}

L^inf norm

\int_X |fg|\,d\mu \le \|f\|_p \|g\|_q, \quad \frac{1}{p}+\frac{1}{q}=1

Holder's inequality

\|f+g\|_p \le \|f\|_p + \|g\|_p

Minkowski's inequality

(L^p)^* \cong L^q, \quad \frac{1}{p}+\frac{1}{q}=1,\; 1 < p < \infty

Duality of L^p

Notation

Notation	Meaning
$L^p(X,\mu)$	Space of p-integrable functions on measure space (X,mu)
$L^p[a,b]$	L^p with Lebesgue measure on [a,b]
$\\|f\\|_p$	L^p norm of f
$p' = p/(p-1)$	Conjugate exponent to p: 1/p + 1/p' = 1

Properties

Inclusions for finite measure spaces

If mu(X) < inf and 1 <= p <= q <= inf, then L^q(X) subset L^p(X) and ||f||_p <= mu(X)^{1/p - 1/q} ||f||_q.

Density of simple functions

For 1 <= p < inf, the simple functions are dense in L^p(X, mu). On R^n, C_c^inf(R^n) is dense in L^p(R^n).

Theorems

Theorem 15.1: Holder's Inequality

Let 1/p + 1/q = 1 with 1 <= p <= infinity. If f in L^p and g in L^q, then fg in L^1 and ||fg||_1 <= ||f||_p ||g||_q. Equality holds iff |f|^p and |g|^q are proportional almost everywhere.

Theorem 15.2: Minkowski's Inequality

For 1 <= p <= infinity and f, g in L^p: ||f + g||_p <= ||f||_p + ||g||_p. This is the triangle inequality for the L^p norm.

Theorem 15.3: Riesz-Fischer Theorem

For every 1 <= p <= infinity, the space L^p(X, mu) is complete: it is a Banach space. Every Cauchy sequence in L^p has a subsequence converging almost everywhere.

Theorem 15.4: Duality of L^p Spaces

For 1 <= p < infinity with conjugate exponent q (1/p + 1/q = 1), every bounded linear functional on L^p is of the form f |-> integral fg d mu for unique g in L^q. The map g |-> integral(·)g is an isometric isomorphism L^q -> (L^p)*.

Worked Examples

1
If ||f||_p = 0 or ||g||_q = 0, both sides are 0. Assume both are positive.
2
Apply Young's inequality: ab <= a^p/p + b^q/q for a, b >= 0.
$ab \le \frac{a^p}{p} + \frac{b^q}{q}, \quad a,b \ge 0$
3
Set a = |f(x)|/||f||_p and b = |g(x)|/||g||_q in Young's inequality and integrate:
$\frac{|f(x)g(x)|}{\|f\|_p\|g\|_q} \le \frac{|f(x)|^p}{p\|f\|_p^p} + \frac{|g(x)|^q}{q\|g\|_q^q}$
4
Integrating both sides: integral (|fg|)/(||f||_p ||g||_q) d mu <= (1/p) integral |f|^p/||f||_p^p + (1/q) integral |g|^q/||g||_q^q = 1/p + 1/q = 1.
$\int \frac{|fg|}{\|f\|_p\|g\|_q}\,d\mu \le \frac{1}{p} + \frac{1}{q} = 1$
5
Multiplying through by ||f||_p ||g||_q gives Holder's inequality.

✓ Answer

Holder's inequality follows from Young's inequality ab <= a^p/p + b^q/q after normalising f and g.

Practice Problems

Mediumproof writing

Use the Riesz-Fischer theorem (completeness of L^p) to prove: if sum_{n=1}^inf ||f_n||_p < infinity, then the series sum f_n converges in L^p.

Easyfill in blank

For the L^p space on [0,1] with Lebesgue measure, the conjugate exponent to p=4 is q = ___. Holder's inequality then says integral |fg| <= ___ .

MediumMultiple choice

Which L^p space is also a Hilbert space?

L^1L^2L^4L^inf

Common Mistakes

Common Mistake

Thinking L^p subset L^q always when p < q.

The inclusion L^q subset L^p holds (in the sense ||f||_p <= C||f||_q) only for finite measure spaces. On R with Lebesgue measure, L^1 and L^2 are neither contained in each other.

Common Mistake

Confusing Holder's inequality with the Cauchy-Schwarz inequality.

Cauchy-Schwarz is Holder's inequality with p = q = 2: integral |fg| <= ||f||_2 ||g||_2. Holder generalises to all conjugate pairs p, q with 1/p + 1/q = 1.

Quiz

Historical Background

The L^2 space of square-integrable functions was used implicitly by Fourier and explicitly by Hilbert in his study of integral equations (1906). Frigyes Riesz introduced the L^p spaces for general p in 1910, proved the Riesz-Fischer theorem (completeness of L^2), and established the duality (L^p)* = L^q. Ernst Fischer proved completeness of L^2 independently. The generalisation to arbitrary measure spaces was systematised by Lebesgue and his school.

1906
Hilbert studies the l^2 sequence space and integral operators
David Hilbert
1907
Riesz and Fischer prove the Riesz-Fischer theorem (completeness of L^2)
Frédéric Riesz, Ernst Fischer
1910
Riesz introduces L^p spaces for general p >= 1
Frédéric Riesz
1936
Clarkson proves uniform convexity and reflexivity of L^p (1 < p < inf)
James Clarkson

Summary

L^p(X,mu) is the space of (equivalence classes of) measurable functions with ||f||_p = (integral |f|^p)^{1/p} < infinity.
L^p is a Banach space for every 1 <= p <= infinity (Riesz-Fischer theorem). L^2 is the unique Hilbert space among them.
Holder's inequality: integral |fg| <= ||f||_p ||g||_q for conjugate exponents 1/p + 1/q = 1.
Minkowski's inequality: ||f+g||_p <= ||f||_p + ||g||_p (triangle inequality for the L^p norm).
Duality: (L^p)* = L^q for 1 < p < inf. L^p is reflexive for 1 < p < inf; L^1 and L^inf are not reflexive.

References

BookRudin, W. — Real and Complex Analysis (3rd ed.), Chapter 3
BookBrezis, H. — Functional Analysis, Sobolev Spaces and PDEs, Chapter 4
WebsiteWikipedia: Lp space