Lp Spaces, Holder and Minkowski Inequalities

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 μ)^{1/ p}

. Hölder's inequality (

∥ f g ∥_{1} \leq ∥ f ∥_{p} ∥ g ∥_{q}

for conjugate exponents

1/ p + 1/ q = 1

) and Minkowski's inequality (the triangle inequality

∥ f + g ∥_{p} \leq ∥ 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).

For a measure space $(X, A, μ)$ and $1 \leq p < \infty$ , the space $L^{p} (μ)$ consists of equivalence classes (modulo $μ$ -a.e. equality) of measurable functions $f$ with $\int_{X} ∣ f ∣^{p} d μ < \infty$ , normed by $∥ f ∥_{p} = (\int ∣ f ∣^{p} d μ)^{1/ p}$ ; the case $p = \infty$ uses the essential supremum. These spaces are the canonical examples of Banach spaces, and $L^{2}$ is the prototypical Hilbert space — the natural setting for Fourier analysis and quantum mechanics. Hölder and Minkowski inequalities are not mere technical lemmas: Minkowski is the statement that $∥ \cdot ∥_{p}$ is a genuine norm, and Hölder underpins duality, interpolation, and the very definition of dual exponents. A surprising fact is that for $0 < p < 1$ the "norm" fails the triangle inequality and $L^{p}$ is a non-locally-convex topological vector space with trivial dual — the geometry collapses entirely below $p = 1$ .

The Core Insight

The entire theory rests on one elementary convexity fact: Young's inequality $ab \leq \frac{a ^{p}}{p} + \frac{b ^{q}}{q}$ , 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.

Formal Statement

Lp Norm and Lp Space

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

∥ f ∥_{p} = (\int_{X} ∣ f ∣^{p} d μ)^{1/ p} .

⟳the exponent

1/ p

is chosen so that

∥ λ f ∥_{p} = ∣ λ ∣ ∥ f ∥_{p}

— i.e. positive homogeneity of degree one.

The space $L^{p} (X, μ)$ is the set of (equivalence classes a.e. of) measurable $f$ with $∥ f ∥_{p} < \infty$ . For $p = \infty$ define the essential supremum

∥ f ∥_{\infty} = x ess sup ∣ f (x) ∣ = in f {M \geq 0 : μ ({∣ f ∣ > M}) = 0} .

⟳take the smallest

M

outside which

∣ f ∣

never exceeds

M

except on a null set; the limit

∥ f ∥_{p} \to ∥ f ∥_{\infty}

p \to \infty

motivates the notation.

Here $A$ is the $σ$ -algebra, $μ$ the measure, and equivalence is $f \sim g ⟺ f = g$ $μ$ -a.e. (necessary so that $∥ f ∥_{p} = 0 \Rightarrow f = 0$ as a class).

Conjugate Exponents

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

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

⟳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.

Young's Inequality (scalar)

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

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

⟳apply concavity of

ln

to the convex combination

\frac{1}{p} ln a^{p} + \frac{1}{q} ln b^{q} \leq ln (\frac{a ^{p}}{p} + \frac{b ^{q}}{q})

with equality iff $a^{p} = b^{q}$ .

Hölder's Inequality

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

∥ f g ∥_{1} = \int_{X} ∣ f g ∣ d μ \leq ∥ f ∥_{p} ∥ g ∥_{q} .

⟳normalize

f, g

to unit norm, apply Young pointwise, integrate.

In particular $f g \in L^{1}$ 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.

Minkowski's Inequality

Let $1 \leq p \leq \infty$ . For $f, g \in L^{p}$ ,

∥ f + g ∥_{p} \leq ∥ f ∥_{p} + ∥ g ∥_{p} .

⟳split

∣ f + g ∣^{p} = ∣ f + g ∣ ∣ f + g ∣^{p - 1}

, bound

∣ f + g ∣ \leq ∣ 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.

Riesz–Fischer Completeness

For every $1 \leq p \leq \infty$ , $L^{p} (X, μ)$ is a Banach space: every Cauchy sequence converges in $L^{p}$ -norm. For $p = 2$ it is a Hilbert space with inner product $⟨ f, g ⟩ = \int f \overset{g}{ˉ} d μ$ .

Why It Holds / Derivation

Young ⟹ Hölder

Step 1. If

∥ f ∥_{p} = 0

∥ g ∥_{q} = 0

then

f g = 0

a.e. and [5] is trivial. If

∥ f ∥_{p} = \infty

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

Step 4. Integrate over

X

\int_{X} F G d μ \leq \frac{1}{p} \int F^{p} + \frac{1}{q} \int G^{q} = \frac{1}{p} + \frac{1}{q} = 1.

Step 5. Multiply through by

∥ f ∥_{p} ∥ g ∥_{q}

\int ∣ f g ∣ d μ \leq ∥ f ∥_{p} ∥ g ∥_{q} .

Endpoint $p = 1, q = \infty$ :

∣ f g ∣ \leq ∣ f ∣ ∥ g ∥_{\infty}

a.e., integrate to get

\int ∣ f g ∣ \leq ∥ g ∥_{\infty} \int ∣ f ∣

■

Hölder ⟹ Minkowski

Step 1. For

p = 1

this is just

∣ f + g ∣ \leq ∣ f ∣ + ∣ g ∣

integrated. For

p = \infty

it follows from

∣ f + g ∣ \leq ∣ f ∣ + ∣ g ∣

a.e. Assume

1 < p < \infty

. Step 2. Note

∥ f + g ∥_{p} < \infty

because

∣ f + g ∣^{p} \leq 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} \leq \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} \leq (∥ f ∥_{p} + ∥ g ∥_{p}) ∥ 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 \frac{1}{p} = 1

∥ f + g ∥_{p} = ∥ f + g ∥_{p}^{p - p / q} \leq ∥ f ∥_{p} + ∥ g ∥_{p} . ■

Young's inequality: the area $ab$ is bounded by the two integral regions under $y=x^{p-1}$

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 ( $∣ f (x) - f (y) ∣ \leq C ∣ x - y ∣^{α}$ ) 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.

Building 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 ${∥ f ∥_{p} \leq 1}$ 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.

Worked Examples

Hölder on a finite interval

Given:

f (x) = x

g (x) = x^{2}

X = [0, 1]

with Lebesgue measure. Use

p = q = 2

(Cauchy–Schwarz). Step 1. Left side:

\int_{0}^{1} ∣ f g ∣ = \int_{0}^{1} x^{3} d x = \frac{1}{4} = 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 \leq 0.2582

. ✓ The inequality holds and is nearly tight because

x

and

x^{2}

are positively correlated but not proportional.

\int_{0}^{1} x^{3} d x = 0.25 \leq 0.2582 = ∥ f ∥_{2} ∥ g ∥_{2}

Minkowski in a sequence space

Given: In

ℓ^{3} = L^{3} (N, 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} = 1 0^{1/3} \approx 2.1544.

Step 4. Right-hand side:

∥ a ∥_{3} + ∥ b ∥_{3} \approx 2.5198.

Step 5. Check

2.1544 \leq 2.5198.

✓ Triangle inequality satisfied, with slack because

a, b

are not parallel.

∥ a + b ∥_{3} = 1 0^{1/3} \approx 2.1544 \leq 2.5198 = ∥ a ∥_{3} + ∥ b ∥_{3}

Common Misconceptions

$∥ \cdot ∥_{p}$ is a norm only for $p \geq 1$ . 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 μ

(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

[1, \infty)

lies in

L^{2}

but not

L^{1}

. Inclusion

L^{p} \subset L^{q}

(

p > q

) holds only on finite measure spaces. 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 \neq \leq 2

); the inequality in fact reverses. This is why

L^{1/2}

is not a normed space and why Minkowski genuinely requires

p \geq 1

Connections

L^{p}

theory threads through nearly all of analysis. The duality

(L^{p})^{*} ≅ 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 - θ}{p _{0}} + \frac{θ}{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.

Real-World Applications

Quantum mechanics. Wavefunctions live in $L^{2} (R^{3})$ ; the normalization $∥ ψ ∥_{2} = 1$ encodes total probability one, and Cauchy–Schwarz underlies the Heisenberg uncertainty principle.

Signal processing & Fourier analysis. The Hausdorff–Young inequality $∥ \hat{f} ∥_{q} \leq ∥ f ∥_{p}$ ( $1 \leq p \leq 2$ ) and Plancherel ( $L^{2}$ isometry of the Fourier transform) govern energy in frequency domains.

Image and data compression. $L^{p}$ norms (and their cousins, $ℓ^{1}$ for sparsity) quantify reconstruction error; minimizing $ℓ^{1}$ subject to constraints is the heart of compressed sensing.

Partial differential equations. Sobolev spaces $W^{k, p}$ are built on $L^{p}$ ; energy estimates and existence theorems (e.g. for Navier–Stokes, elliptic regularity) hinge on Hölder and Sobolev embeddings.

Statistics & machine learning. $L^{2}$ (ridge) and $L^{1}$ (LASSO) regularization correspond to different unit balls; Hölder bounds appear in concentration and generalization estimates.

Numerical analysis. Error norms $∥ u - u_{h} ∥_{p}$ measure finite-element approximation quality; Hölder converts pointwise estimates into integral ones.

Economics & optimization. Hölder/Cauchy–Schwarz bound inner products of price and quantity vectors; duality in $L^{p}$ underlies linear programming in infinite dimensions.

Historical Development

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.

Subtleties & Limits

Several delicate boundary phenomena persist. At $p = 1$ and $p = \infty$ the spaces are non-reflexive: $(L^{1})^{*} = L^{\infty}$ but $(L^{\infty})^{*} ⊋ L^{1}$ (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

The Core Insight

Formal Statement

Lp Norm and Lp Space

Conjugate Exponents

Young's Inequality (scalar)

Hölder's Inequality

Minkowski's Inequality

Riesz–Fischer Completeness

Why It Holds / Derivation

Young ⟹ Hölder

Hölder ⟹ Minkowski

Young's inequality: the area $ab$ is bounded by the two integral regions under $y=x^{p-1}$

Otto Hölder | 1859–1937

Building Intuition

Worked Examples

Hölder on a finite interval

Minkowski in a sequence space

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads