Lp Spaces, Holder and Minkowski Inequalities

⚡ Key Takeaways

Lp spaces are normed vector spaces of measurable functions whose $p$ -th power is integrable, equipped with the norm $\|f\|_p = (\int |f|^p \, d\mu)^{1/p}$ . Hölder's inequality ( $\|fg\|_1 \le \|f\|_p \|g\|_q$ for conjugate exponents $1/p+1/q=1$ ) and Minkowski's inequality (the triangle inequality $\|f+g\|_p \le \|f\|_p + \|g\|_p$ ) are the two pillars that make $L^p$ a complete normed space. Applies wherever integration, probability, PDE, harmonic analysis, or functional analysis live. Hölder (1889), Minkowski (1896), Riesz–Fischer completeness (1907).

💡 The Core Insight

The entire theory rests on one elementary convexity fact: Young's inequality $ab \le \frac{a^p}{p} + \frac{b^q}{q}$ , itself a restatement of the concavity of the logarithm. From this single seed grow Hölder (by integrating a normalized product), Minkowski (by writing $|f+g|^p = |f+g|\,|f+g|^{p-1}$ and applying Hölder), and ultimately the whole duality theory $(L^p)^* = L^q$ . The numbers $p$ and $q$ with $1/p + 1/q = 1$ are forced by the scaling that makes Young's inequality sharp.

DefinitionLp Norm and Lp Space

Let $(X, \mathcal{A}, \mu)$ be a measure space and $0 < p < \infty$ . For a measurable $f: X \to \mathbb{C}$ (or $\mathbb{R}$ ) define

1

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

Derivation hint: the exponent $1/p$ is chosen so that $\|\lambda f\|_p = |\lambda|\,\|f\|_p$ — i.e. positive homogeneity of degree one. The space $L^p(X, \mu)$ is the set of (equivalence classes a.e. of) measurable $f$ with $\|f\|_p < \infty$ . For $p = \infty$ define the essential supremum

2

\quad \|f\|_\infty = \operatorname*{ess\,sup}_{x} |f(x)| = \inf\{ M \ge 0 : \mu(\{|f| > M\}) = 0 \}.

Derivation hint: take the smallest $M$ outside which $|f|$ never exceeds $M$ except on a null set; the limit $\|f\|_p \to \|f\|_\infty$ as $p\to\infty$ motivates the notation. Here $\mathcal{A}$ is the $\sigma$ -algebra, $\mu$ the measure, and equivalence is $f \sim g \iff f = g$ $\mu$ -a.e. (necessary so that $\|f\|_p = 0 \Rightarrow f = 0$ as a class).

DefinitionConjugate Exponents

Two numbers $p, q \in [1, \infty]$ are conjugate (or Hölder conjugate) if

3

\quad \frac{1}{p} + \frac{1}{q} = 1,

Derivation hint: convention $1/\infty = 0$ , so $p=1$ pairs with $q=\infty$ and $p=2$ is self-conjugate. equivalently $q = p/(p-1)$ . The self-conjugate point $p=q=2$ is exactly where $L^p$ becomes a Hilbert space.

TheoremYoung's Inequality (scalar)

For $a, b \ge 0$ and conjugate exponents $1 < p, q < \infty$ ,

4

\quad ab \le \frac{a^p}{p} + \frac{b^q}{q},

Derivation hint: apply concavity of $\ln$ to the convex combination $\frac1p \ln a^p + \frac1q \ln b^q \le \ln(\frac{a^p}{p} + \frac{b^q}{q})$ . with equality iff $a^p = b^q$ .

TheoremHölder's Inequality

Let $1 \le p, q \le \infty$ be conjugate. For measurable $f, g$ ,

5

\quad \|fg\|_1 = \int_X |fg| \, d\mu \le \|f\|_p \, \|g\|_q.

Derivation hint: normalize $f, g$ to unit norm, apply Young pointwise, integrate. In particular $fg \in L^1$ whenever $f \in L^p$ , $g \in L^q$ . Equality holds iff $|f|^p$ and $|g|^q$ are proportional a.e. (for $1<p<\infty$ ). The case $p=q=2$ is the Cauchy–Schwarz inequality.

TheoremMinkowski's Inequality

Let $1 \le p \le \infty$ . For $f, g \in L^p$ ,

6

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

Derivation hint: split $|f+g|^p = |f+g|\,|f+g|^{p-1}$ , bound $|f+g|\le|f|+|g|$ , then apply Hölder with exponents $p$ and $q=p/(p-1)$ to each term. This is precisely the triangle inequality, so $\|\cdot\|_p$ is a norm and $L^p$ a normed vector space.

TheoremRiesz–Fischer Completeness

For every $1 \le p \le \infty$ , $L^p(X,\mu)$ is a Banach space: every Cauchy sequence converges in $L^p$ -norm. For $p=2$ it is a Hilbert space with inner product $\langle f, g\rangle = \int f \bar g \, d\mu$ .

ProofYoung ⟹ Hölder

Step 1. If $\|f\|_p = 0$ or $\|g\|_q = 0$ then $fg = 0$ a.e. and [5] is trivial. If $\|f\|_p = \infty$ or $\|g\|_q = \infty$ the right side is $\infty$ and [5] holds. So assume $0 < \|f\|_p, \|g\|_q < \infty$ .

Step 2. Define normalized functions $F = |f|/\|f\|_p$ and $G = |g|/\|g\|_q$ , so $\int F^p = \int G^q = 1$ .

Step 3. Apply Young's inequality [4] pointwise with $a = F(x)$ , $b = G(x)$ :

F(x) G(x) \le \frac{F(x)^p}{p} + \frac{G(x)^q}{q}.

Step 4. Integrate over $X$ :

\int_X FG \, d\mu \le \frac1p \int F^p + \frac1q \int G^q = \frac1p + \frac1q = 1.

Step 5. Multiply through by $\|f\|_p\|g\|_q$ :

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

Endpoint $p=1, q=\infty$ : $|fg| \le |f|\,\|g\|_\infty$ a.e., integrate to get $\int|fg| \le \|g\|_\infty \int |f|$ . $\blacksquare$

∎

ProofHölder ⟹ Minkowski

Step 1. For $p=1$ this is just $|f+g| \le |f| + |g|$ integrated. For $p=\infty$ it follows from $|f+g| \le |f|+|g|$ a.e. Assume $1 < p < \infty$ .

Step 2. Note $\|f+g\|_p < \infty$ because $|f+g|^p \le 2^{p-1}(|f|^p + |g|^p)$ (convexity of $t \mapsto t^p$ ). If $\|f+g\|_p = 0$ , done.

Step 3. Write

\|f+g\|_p^p = \int |f+g|^p = \int |f+g|\,|f+g|^{p-1} \le \int |f|\,|f+g|^{p-1} + \int |g|\,|f+g|^{p-1}.

Step 4. Apply Hölder to each term with exponents $p$ and $q = p/(p-1)$ . Note $(p-1)q = p$ , so $\| \,|f+g|^{p-1}\, \|_q = (\int |f+g|^{(p-1)q})^{1/q} = \|f+g\|_p^{p/q}$ :

\|f+g\|_p^p \le \big(\|f\|_p + \|g\|_p\big)\,\|f+g\|_p^{p/q}.

Step 5. Divide by $\|f+g\|_p^{p/q}$ (finite and nonzero). Since $p - p/q = p(1 - 1/q) = p \cdot \frac1p = 1$ :

\|f+g\|_p = \|f+g\|_p^{\,p - p/q} \le \|f\|_p + \|g\|_p. \qquad \blacksquare

∎

Figure

Young's inequality: the area

ab

is bounded by the two integral regions under

y=x^{p-1}

mathematician · Otto Hölder | 1859–1937

A German mathematician at Tübingen and Leipzig, Hölder published the inequality bearing his name in 1889 — though the discrete case was found earlier by L. J. Rogers (1888), so it is sometimes called the Rogers–Hölder inequality. Hölder's broader fame rests on group theory: the Jordan–Hölder theorem on composition series and the classification of finite simple groups of small order. His work on Hölder continuity ( $|f(x)-f(y)| \le C|x-y|^\alpha$ ) and Hölder summability of series further cemented his name throughout analysis. The inequality emerged from his study of convexity and mean values, the same circle of ideas that Jensen and Minkowski were independently developing.

🧠 Intuition

Think of Hölder as a weighted accounting of mass. The exponent $p$ measures how much you penalize large values: large $p$ emphasizes peaks (the sup), small $p$ emphasizes spread. Conjugacy $1/p + 1/q = 1$ is the exact trade under which a "tall-and-thin" function $f$ (large $p$ ) can be paired with a "short-and-wide" function $g$ (small $q$ ) so that their product's integral stays controlled. Cauchy–Schwarz ( $p=q=2$ ) is the symmetric, "fair" balance point. Minkowski says vectors in $L^p$ obey the ordinary triangle inequality — the unit ball $\{\|f\|_p \le 1\}$ is convex. As $p$ increases from $1$ to $\infty$ this ball morphs from a diamond ( $p=1$ ) through a round disk ( $p=2$ ) toward a square ( $p=\infty$ ). Convexity is exactly what fails for $0<p<1$ : there the ball becomes a "pinched star" (astroid-like) and adding two unit vectors can produce something of norm $>2$ . That single geometric defect is why $L^p$ for $p<1$ has no nontrivial continuous linear functionals.

ExampleHölder on a finite interval

Given: $f(x) = x$ , $g(x) = x^2$ on $X = [0,1]$ with Lebesgue measure. Use $p = q = 2$ (Cauchy–Schwarz).

Step 1. Left side: $\int_0^1 |fg| = \int_0^1 x^3 \, dx = \tfrac14 = 0.25.$

Step 2. $\|f\|_2 = (\int_0^1 x^2)^{1/2} = (1/3)^{1/2} \approx 0.5774.$

Step 3. $\|g\|_2 = (\int_0^1 x^4)^{1/2} = (1/5)^{1/2} \approx 0.4472.$

Step 4. Product of norms: $0.5774 \times 0.4472 \approx 0.2582.$

Step 5. Check $0.25 \le 0.2582$ . ✓ The inequality holds and is nearly tight because $x$ and $x^2$ are positively correlated but not proportional.

\boxed{\int_0^1 x^3\,dx = 0.25 \;\le\; 0.2582 = \|f\|_2\|g\|_2}

ExampleMinkowski in a sequence space

Given: In $\ell^3 = L^3(\mathbb{N}, \text{counting measure})$ , take $a = (1, 1, 0, 0, \dots)$ and $b = (0, 1, 1, 0, \dots)$ . Verify Minkowski.

Step 1. $\|a\|_3 = (1^3 + 1^3)^{1/3} = 2^{1/3} \approx 1.2599.$ Same for $\|b\|_3 \approx 1.2599.$

Step 2. Sum: $a + b = (1, 2, 1, 0, \dots).$

Step 3. $\|a+b\|_3 = (1^3 + 2^3 + 1^3)^{1/3} = (1 + 8 + 1)^{1/3} = 10^{1/3} \approx 2.1544.$

Step 4. Right-hand side: $\|a\|_3 + \|b\|_3 \approx 2.5198.$

Step 5. Check $2.1544 \le 2.5198.$ ✓ Triangle inequality satisfied, with slack because $a, b$ are not parallel.

\boxed{\|a+b\|_3 = 10^{1/3} \approx 2.1544 \;\le\; 2.5198 = \|a\|_3 + \|b\|_3}

⚠️ Common mistake

$\|\cdot\|_p$ is a norm only for $p \ge 1$ . For $0 < p < 1$ , $\|\cdot\|_p$ fails the triangle inequality, so it is not a norm — though $d(f,g) = \int|f-g|^p \,d\mu$ (without the $1/p$ power) is a metric. Also beware: elements of $L^p$ are equivalence classes, not functions; " $f(x_0)$ " is meaningless for a single point under Lebesgue measure. Finally, on a space of infinite measure there is no inclusion between $L^p$ spaces — e.g. $1/x$ on $[1,\infty)$ lies in $L^2$ but not $L^1$ . Inclusion $L^p \subset L^q$ ( $p>q$ ) holds only on finite measure spaces.

Counterexample

Failure of Minkowski for $p = 1/2$ . Take $X = \{1, 2\}$ with counting measure, $f = (1, 0)$ , $g = (0, 1)$ , and "norm" $\|h\|_{1/2} = (\sum |h_i|^{1/2})^2$ . Then $\|f\|_{1/2} = (1)^2 = 1$ and $\|g\|_{1/2} = 1$ , so the right side is $2$ . But $f + g = (1,1)$ gives

\|f+g\|_{1/2} = (1^{1/2} + 1^{1/2})^2 = 2^2 = 4 > 2.

The triangle inequality is violated ( $4 \not\le 2$ ); the inequality in fact reverses. This is why $L^{1/2}$ is not a normed space and why Minkowski genuinely requires $p \ge 1$ .

connection

$L^p$ theory threads through nearly all of analysis. The duality $(L^p)^* \cong L^q$ (via Riesz Representation Theorem) is the prototype of Dual Spaces and is proved directly using Hölder for the upper bound. Hilbert Spaces are exactly $L^2$ in disguise, where Cauchy–Schwarz is the special self-conjugate case. Interpolation Theory (Riesz–Thorin, Marcinkiewicz) extends bounds known at two exponents $p_0, p_1$ to all intermediate $p$ , using a generalized Hölder identity $\frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ . In probability, $L^p$ norms are the $p$ -th moments and Hölder yields covariance bounds; Jensen's Inequality is the convexity statement dual to these. The reflexivity of $L^p$ for $1 < p < \infty$ (and its failure at $p = 1, \infty$ ) is foundational in Banach Space Geometry.

🔬 Where this appears

⏳ History

- 1859 — Cauchy & Bunyakovsky / Schwarz. The discrete Cauchy inequality (1821) is extended to integrals by Bunyakovsky (1859) and rediscovered by H. A. Schwarz (1888) — the

p=q=2

ancestor.

- 1888 — L. J. Rogers. Publishes the discrete (sequence) form of what is now Hölder's inequality, predating Hölder.

- 1889 — Otto Hölder. Proves the inequality in his paper on mean values, building on Jensen-type convexity, giving the general conjugate-exponent form.

- 1896 — Hermann Minkowski. In Geometrie der Zahlen establishes the triangle inequality for

L^p

-type sums, in the context of convex bodies and number theory.

- 1902 — Henri Lebesgue. Introduces the Lebesgue integral, providing the measure-theoretic foundation that makes

L^p

spaces complete.

- 1906–1910 — Fréchet & Riesz. Maurice Fréchet abstracts metric spaces; Frigyes Riesz systematically develops

L^p

spaces as function spaces.

- 1907 — Riesz–Fischer theorem. F. Riesz and E. Fischer independently prove completeness of

L^2

, securing it as a Hilbert space; soon generalized to all

L^p

.

- 1910 — F. Riesz. Proves the duality

(L^p)^* = L^q

for

1 < p < \infty

, cementing Hölder's central role.

- 1922–1932 — Banach. Stefan Banach's Théorie des opérations linéaires places

L^p

as the central example of Banach spaces, founding modern functional analysis.

open-problem

Several delicate boundary phenomena persist. At $p=1$ and $p=\infty$ the spaces are non-reflexive: $(L^1)^* = L^\infty$ but $(L^\infty)^* \supsetneq L^1$ (the dual of $L^\infty$ involves finitely-additive measures, requiring the axiom of choice to exhibit). For $0<p<1$ , $L^p[0,1]$ is a complete metric F-space with trivial dual $(L^p)^* = \{0\}$ — a striking failure of the Hahn–Banach machinery. The structure of $L^\infty$ and its dual remains subtle (it is a commutative von Neumann algebra / $C^*$ -algebra, tied to deep questions in operator algebras). Sharp constants in Hölder-type and Hausdorff–Young inequalities (Beckner's constants, the Babenko–Beckner inequality) were only resolved in the 1970s, and optimal constants in many functional inequalities (e.g. sharp Sobolev, Gagliardo–Nirenberg) remain active research. Whether certain $L^p$ bounds extend to operators on general measure spaces ("UMD" property, Kato's square root problem — solved 2002) shows how these classical inequalities still drive frontier analysis.

Lp Spaces, Holder and Minkowski Inequalities

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also