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RA-9

8-13 hours

Advanced

Lp Spaces & Duality

Master Lp spaces, Hölder and Minkowski inequalities, completeness, and duality theory. Understand the Riesz representation theorem and the special role of L2 as a Hilbert space.

Learning Objectives

By the end of this course, you will be able to:

Understand Lp spaces and their norms for 1 ≤ p ≤ ∞

Master Hölder's and Minkowski's inequalities

Prove that Lp spaces are complete (Banach spaces)

Understand the dual space of Lp and the Riesz representation theorem

Apply Lp space theory to solve problems

Understand the special case of L2 as a Hilbert space

Prerequisites

Before starting this course, you should be familiar with:

Lebesgue integration and L1 space
Convergence theorems
Basic functional analysis (norms, Banach spaces)
Measure theory

Core Concepts

Lp spaces and duality theory

Definition 9.1: Lp Spaces

For $1 \leq p < \infty$ , the space $L^p(X, \mu)$ consists of all measurable functions $f$ such that:

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

The space $L^\infty(X, \mu)$ consists of functions with:

\|f\|_\infty = \text{ess sup}_{x \in X} |f(x)| < \infty

Functions that are equal almost everywhere are identified.

Definition 9.2: Conjugate Exponents

Two numbers $p$ and $q$ with $1 < p, q < \infty$ are called conjugate exponents if:

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

We also say $(1, \infty)$ and $(\infty, 1)$ are conjugate pairs.

Theorem 9.1: Hölder's Inequality

Let $p$ and $q$ be conjugate exponents with $1 \leq p, q \leq \infty$ . If $f \in L^p$ and $g \in L^q$ , then $fg \in L^1$ and:

\|fg\|_1 \leq \|f\|_p \|g\|_q

Equality holds if and only if $|f|^p$ and $|g|^q$ are proportional almost everywhere.

Proof of Theorem 9.1:

Proof:

For $p = 1, q = \infty$ , the result is immediate. For $1 < p < \infty$ , use Young's inequality:

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

Apply this to $a = |f|/\|f\|_p$ and $b = |g|/\|g\|_q$ , then integrate. ∎

∎

Theorem 9.2: Minkowski's Inequality

For $1 \leq p \leq \infty$ and $f, g \in L^p$ :

\|f + g\|_p \leq \|f\|_p + \|g\|_p

This is the triangle inequality for Lp spaces.

Proof of Theorem 9.2:

Proof:

For $p = 1$ or $p = \infty$ , the result is straightforward.

For $1 < p < \infty$ , write $|f + g|^p = |f + g| |f + g|^{p-1}$ and apply Hölder's inequality to each term, then use that $(p-1)q = p$ . ∎

∎

Theorem 9.3: Riesz-Fischer Theorem

For $1 \leq p \leq \infty$ , the space $L^p$ is complete, i.e., it is a Banach space.

Proof of Theorem 9.3:

Proof:

Let $\{f_n\}$ be a Cauchy sequence in $L^p$ . Extract a subsequence $\{f_{n_k}\}$ such that $\|f_{n_{k+1}} - f_{n_k}\|_p < 2^{-k}$ .

Define $g = |f_{n_1}| + \sum_{k=1}^\infty |f_{n_{k+1}} - f_{n_k}|$ . By Minkowski, $g \in L^p$ , so the series converges a.e.

The limit $f = \lim f_{n_k}$ exists a.e. and is in $L^p$ . The full sequence converges to $f$ by the Cauchy property. ∎

∎

Theorem 9.4: Riesz Representation Theorem

Let $1 < p < \infty$ and let $q$ be the conjugate exponent. Then every bounded linear functional $\phi$ on $L^p$ is of the form:

\phi(f) = \int fg \, d\mu

for a unique $g \in L^q$ . Moreover, $\|\phi\| = \|g\|_q$ .

Proof of Theorem 9.4:

Proof:

For finite measure spaces, define a set function $\nu(E) = \phi(\chi_E)$ . Show that $\nu \ll \mu$ and apply Radon-Nikodym to get $g$ .

Extend to σ-finite spaces by taking limits. The isometry follows from Hölder's inequality. ∎

∎

Example 9.1: Computing an Lp Norm

Problem: Compute $\|f\|_p$ where $f(x) = x^{-1/2}$ on $(0,1]$ .

Solution:

We have $\|f\|_p^p = \int_0^1 x^{-p/2} \, dx$ .

This converges if and only if $-p/2 > -1$ , i.e., $p < 2$ .

For $p < 2$ , $\|f\|_p = \left(\frac{2}{2-p}\right)^{1/p}$ .

For $p \geq 2$ , $f \notin L^p$ .

Corollary 9.1: L2 is a Hilbert Space

The space $L^2$ is a Hilbert space with inner product:

\langle f, g \rangle = \int fg \, d\mu

The norm is induced by this inner product: $\|f\|_2 = \sqrt{\langle f, f \rangle}$ .

Theorem 9.5: Riesz-Fischer Theorem (Complete Proof)

For $1 \leq p < \infty$ , the space $L^p(\mu)$ is complete with respect to the $L^p$ norm.

That is, every Cauchy sequence in $L^p$ converges to a function in $L^p$ .

Proof of Theorem 9.5:

Proof:

Let $\{f_n\}$ be a Cauchy sequence in $L^p$ . Extract a subsequence $\{f_{n_k}\}$ such that $\|f_{n_{k+1}} - f_{n_k}\|_p < 2^{-k}$ .

Define $g = |f_{n_1}| + \sum_{k=1}^\infty |f_{n_{k+1}} - f_{n_k}|$ . By Minkowski's inequality,

\|g\|_p \leq \|f_{n_1}\|_p + \sum_{k=1}^\infty 2^{-k} < \infty

so $g \in L^p$ . This implies the series converges almost everywhere, so $f = \lim f_{n_k}$ exists a.e.

By Fatou's lemma, $\|f - f_n\|_p \leq \liminf_{k \to \infty} \|f_{n_k} - f_n\|_p$ , which tends to 0 as $n \to \infty$ by the Cauchy property. ∎

∎

Theorem 9.6: Riesz Representation Theorem (Detailed Proof)

Let $1 < p < \infty$ and $q$ be the conjugate exponent. Every bounded linear functional $\phi: L^p \to \mathbb{R}$ has the form

\phi(f) = \int fg \, d\mu

for a unique $g \in L^q$ , with $\|\phi\| = \|g\|_q$ .

Proof of Theorem 9.6:

Proof:

Step 1: For a finite measure space, define $\nu(E) = \phi(\chi_E)$ for measurable sets $E$ .

Step 2: Show that $\nu$ is a signed measure with $\nu \ll \mu$ (using the boundedness of $\phi$ ).

Step 3: Apply the Radon-Nikodym theorem to get $g \in L^1$ such that $\nu(E) = \int_E g \, d\mu$ .

Step 4: Show that $g \in L^q$ by testing $\phi$ on functions of the form $f = |g|^{q-1} \text{sgn}(g)$ .

Step 5: Extend to simple functions, then to all of $L^p$ by density.

Step 6: The isometry follows from Hölder's inequality and the choice of test functions. ∎

∎

Example 9.4: Computing Lp Norm: Detailed Steps

Problem: Compute $\|f\|_p$ where $f(x) = x^{-\alpha}$ on $(0,1]$ for $\alpha > 0$ .

Solution:

We have $\|f\|_p^p = \int_0^1 x^{-\alpha p} \, dx$ .

This integral converges if and only if $-\alpha p > -1$ , i.e., $\alpha p < 1$ or $p < 1/\alpha$ .

For $p < 1/\alpha$ ,

\|f\|_p^p = \int_0^1 x^{-\alpha p} \, dx = \left[\frac{x^{1-\alpha p}}{1-\alpha p}\right]_0^1 = \frac{1}{1-\alpha p}

Therefore, $\|f\|_p = (1-\alpha p)^{-1/p}$ .

For $p \geq 1/\alpha$ , $f \notin L^p$ .

Example 9.5: L2 as a Hilbert Space: Orthogonality

Problem: Show that the functions $f_n(x) = e^{2\pi i n x}$ on $[0,1]$ form an orthonormal set in $L^2([0,1])$ .

Solution:

For $n \neq m$ ,

\langle f_n, f_m \rangle = \int_0^1 e^{2\pi i n x} \overline{e^{2\pi i m x}} \, dx = \int_0^1 e^{2\pi i (n-m) x} \, dx = 0

For $n = m$ ,

\langle f_n, f_n \rangle = \int_0^1 |e^{2\pi i n x}|^2 \, dx = \int_0^1 1 \, dx = 1

Therefore, $\{f_n\}$ is an orthonormal set. This is the basis for Fourier series theory.

Example 9.6: Constructing Dual Space Elements

Problem: Given $g \in L^q$ where $q$ is the conjugate of $p$ , show that $\phi(f) = \int fg$ defines a bounded linear functional on $L^p$ .

Solution:

By Hölder's inequality,

|\phi(f)| = \left|\int fg\right| \leq \|f\|_p \|g\|_q

so $\phi$ is bounded with $\|\phi\| \leq \|g\|_q$ .

To show equality, take $f = |g|^{q-1} \text{sgn}(g) / \|g\|_q^{q-1}$ . Then $\|f\|_p = 1$ and $\phi(f) = \|g\|_q$ , so $\|\phi\| = \|g\|_q$ .

This shows that every $g \in L^q$ gives rise to a bounded linear functional on $L^p$ .

Example 9.7: Application of Lp Completeness

Problem: Show that if $f_n \to f$ in $L^p$ and $f_n \to g$ pointwise a.e., then $f = g$ a.e.

Solution:

Since $f_n \to f$ in $L^p$ , there is a subsequence $\{f_{n_k}\}$ such that $f_{n_k} \to f$ pointwise a.e.

But we also have $f_n \to g$ pointwise a.e., so $f_{n_k} \to g$ pointwise a.e.

Therefore, $f = g$ almost everywhere.

This demonstrates that Lp limits are unique up to almost everywhere equality, which is a consequence of completeness.

Corollary 9.3: Dual Properties of Lp Spaces

For $1 < p < \infty$ , the dual space $(L^p)^*$ is isometrically isomorphic to $L^q$ where $1/p + 1/q = 1$ .

This means:

Every bounded linear functional on $L^p$ comes from an element of $L^q$
The norm of the functional equals the Lq norm of the representing function
The identification is natural: $g \mapsto (f \mapsto \int fg)$

Remark 9.3: Importance of Riesz Representation Theorem

The Riesz representation theorem is one of the most important results in functional analysis:

Dual spaces: It completely characterizes the dual space of Lp, showing that it is "the same" as Lq (for $1 < p < \infty$ ).
Weak convergence: The theorem is essential for understanding weak convergence in Lp spaces.
Variational problems: Many optimization problems in analysis reduce to finding elements of dual spaces, which the theorem identifies with Lq functions.
PDE theory: The theorem is used constantly in partial differential equations, where solutions are often found as elements of Lp spaces and their duals.
Probability: In stochastic analysis, martingales and other processes are studied using Lp spaces and their duals.

The theorem was proved by Frigyes Riesz in 1907 for $p = 2$ (the Hilbert space case) and later extended to general $p$ .

Remark 9.4: Applications in Partial Differential Equations

Lp spaces and duality are fundamental in PDE theory:

Sobolev spaces: The spaces $W^{k,p}$ (functions with derivatives in Lp) are Banach spaces whose duals can be characterized using Lp duality.
Weak solutions: Many PDEs are solved in a "weak sense" by interpreting derivatives as distributions, which are elements of dual spaces.
Energy methods: The L2 case (Hilbert space) provides energy estimates through the inner product structure.
Regularity theory: Lp estimates (for various $p$ ) are used to prove regularity of solutions to elliptic and parabolic equations.
Existence theorems: The completeness of Lp spaces and compactness arguments are used to prove existence of solutions.

These applications show why Lp spaces are central to modern analysis and PDE theory.

Example 9.8: Hölder's Inequality: Application

Problem: Use Hölder's inequality to show that if $f \in L^p$ and $g \in L^q$ where $1/p + 1/q = 1$ , then $fg \in L^1$ and $\|fg\|_1 \leq \|f\|_p \|g\|_q$ .

Solution:

This is exactly Hölder's inequality. For $p = q = 2$ , this becomes the Cauchy-Schwarz inequality:

\int |fg| \leq \left(\int |f|^2\right)^{1/2} \left(\int |g|^2\right)^{1/2}

Hölder's inequality is one of the most important inequalities in analysis, used constantly in estimates.

Example 9.9: Lp Spaces: Inclusion Relations

Problem: On a finite measure space, show that $L^p \subseteq L^q$ for $1 \leq q < p \leq \infty$ .

Solution:

For $f \in L^p$ , use Hölder's inequality with exponents $p/q$ and $p/(p-q)$ :

\int |f|^q = \int |f|^q \cdot 1 \leq \left(\int |f|^p\right)^{q/p} \left(\int 1^{p/(p-q)}\right)^{(p-q)/p}

Since the measure space is finite, $\int 1 < \infty$ , so $f \in L^q$ .

This shows that on finite measure spaces, higher Lp spaces are contained in lower ones, the opposite of what happens on infinite measure spaces.

Example 9.10: Riesz Representation: Constructing the Functional

Problem: Given $g \in L^q$ where $q$ is the conjugate of $p$ , show that $\phi(f) = \int fg$ defines a bounded linear functional on $L^p$ with norm $\|\phi\| = \|g\|_q$ .

Solution:

By Hölder's inequality, $|\phi(f)| \leq \|f\|_p \|g\|_q$ , so $\|\phi\| \leq \|g\|_q$ .

To show equality, take $f = |g|^{q-1} \text{sgn}(g) / \|g\|_q^{q-1}$ . Then $\|f\|_p = 1$ and $\phi(f) = \|g\|_q$ , so $\|\phi\| = \|g\|_q$ .

This shows that every $g \in L^q$ gives rise to a bounded linear functional on $L^p$ , and the Riesz representation theorem shows that all functionals arise this way.

Theorem 9.7: Clarkson's Inequalities

For $2 \leq p < \infty$ and $f, g \in L^p$ , we have:

\left\|\frac{f + g}{2}\right\|_p^p + \left\|\frac{f - g}{2}\right\|_p^p \leq \frac{1}{2}(\|f\|_p^p + \|g\|_p^p)

These inequalities are used to prove that Lp spaces are uniformly convex for $1 < p < \infty$ .

Proof of Theorem 9.7:

Proof:

The proof uses the convexity of the function $t \mapsto |t|^p$ for $p \geq 2$ .

For $1 < p < 2$ , there are similar but more complicated inequalities. These results are fundamental in the geometry of Banach spaces. ∎

∎

Corollary 9.4: Uniform Convexity of Lp

For $1 < p < \infty$ , the space $L^p$ is uniformly convex: for any $\epsilon > 0$ , there exists $\delta > 0$ such that if $\|f\|_p = \|g\|_p = 1$ and $\|f - g\|_p \geq \epsilon$ , then $\|(f+g)/2\|_p \leq 1 - \delta$ .

This property is crucial for the geometry of Lp spaces and has important consequences for optimization and fixed point theory.

Remark 9.5: Lp Spaces for p = 1 and p = ∞

The cases $p = 1$ and $p = \infty$ are special:

L1: The dual of L1 is $L^\infty$ , but not all functionals on L1 come from L∞ functions (this requires additional structure).
L∞: The dual of L∞ is much larger than L1. It contains "finitely additive measures" that are not countably additive.
Completeness: Both L1 and L∞ are complete, but L∞ is not separable (in general), making it more difficult to work with.

These special cases require different techniques and have different properties than Lp for $1 < p < \infty$ .

Remark 9.6: Interpolation Theory

Interpolation theory studies how properties of operators on Lp spaces vary with $p$ :

Riesz-Thorin interpolation: If an operator is bounded on Lp0 and Lp1, it is bounded on all Lp for $p$ between p0 and p1.
Marcinkiewicz interpolation: A more general result that works with weak-type bounds, crucial for singular integrals.
Applications: Used to prove boundedness of Fourier multipliers, singular integrals, and other operators in harmonic analysis.

Interpolation theory is a powerful tool that allows us to prove results for all Lp spaces by checking only a few values of $p$ .

Example 9.11: Lp Norm: Computing for a Specific Function

Problem: Compute $\|f\|_p$ where $f(x) = x^{-\alpha}$ on $(0,1]$ for $\alpha > 0$ and determine for which $p$ the function is in $L^p$ .

Solution:

We have $\|f\|_p^p = \int_0^1 x^{-\alpha p} \, dx$ .

This converges if and only if $-\alpha p > -1$ , i.e., $\alpha p < 1$ or $p < 1/\alpha$ .

For $p < 1/\alpha$ ,

\|f\|_p = \left(\frac{1}{1-\alpha p}\right)^{1/p}

For $p \geq 1/\alpha$ , $f \notin L^p$ .

This shows that membership in Lp depends on both the function and the value of $p$ .

Example 9.12: Hölder's Inequality: Optimal Constant

Problem: Show that Hölder's inequality is sharp by finding functions that achieve equality.

Solution:

Equality in Hölder's inequality holds when $|f|^p$ and $|g|^q$ are proportional almost everywhere.

For example, on $[0,1]$ , take $f(x) = x^{1/p}$ and $g(x) = x^{1/q}$ . Then

\int_0^1 |fg| = \int_0^1 x = \frac{1}{2} = \left(\int_0^1 x\right)^{1/p} \left(\int_0^1 x\right)^{1/q} = \|f\|_p \|g\|_q

This shows that Hölder's inequality is optimal: the constant 1 cannot be improved.

Example 9.13: L2 Inner Product: Orthogonality

Problem: Show that the functions $\{1, \cos(nx), \sin(nx) : n \in \mathbb{N}\}$ form an orthogonal set in $L^2([-\pi, \pi])$ .

Solution:

Using trigonometric identities, we can show that:

$\int_{-\pi}^{\pi} \cos(mx) \cos(nx) \, dx = 0$ for $m \neq n$
$\int_{-\pi}^{\pi} \sin(mx) \sin(nx) \, dx = 0$ for $m \neq n$
$\int_{-\pi}^{\pi} \cos(mx) \sin(nx) \, dx = 0$ for all $m, n$
$\int_{-\pi}^{\pi} 1 \cdot \cos(nx) \, dx = 0$ and $\int_{-\pi}^{\pi} 1 \cdot \sin(nx) \, dx = 0$

This orthogonality is the foundation of Fourier series theory.

Theorem 9.8: Duality of Lp and Lq

For $1 < p < \infty$ and $q$ the conjugate exponent, the map $g \mapsto \phi_g$ where $\phi_g(f) = \int fg$ is an isometric isomorphism from $L^q$ onto $(L^p)^*$ .

This means that every bounded linear functional on $L^p$ comes from a unique element of $L^q$ , and the norms are equal.

Proof of Theorem 9.8:

Proof:

The Riesz representation theorem (Theorem 9.4) shows that the map is surjective.

Uniqueness follows from the fact that if $\int fg_1 = \int fg_2$ for all $f \in L^p$ , then $g_1 = g_2$ a.e.

The isometry property $\|\phi_g\| = \|g\|_q$ was shown in Example 9.6. ∎

∎

Corollary 9.5: Reflexivity of Lp

For $1 < p < \infty$ , the space $L^p$ is reflexive: $(L^p)^{**} = L^p$ .

This follows from the duality $(L^p)^* = L^q$ and $(L^q)^* = L^p$ (since $q$ is the conjugate of $p$ and vice versa).

Remark 9.7: Lp Spaces and Fourier Analysis

Lp spaces are central to Fourier analysis:

L2 theory: The Fourier transform is an isometry on L2 (Plancherel theorem), making L2 the natural setting for Fourier analysis.
L1 theory: The Fourier transform maps L1 to a space of continuous functions, but not onto L1.
Lp theory: For $1 < p < 2$ , the Hausdorff-Young inequality gives bounds on the Fourier transform in Lp' where $1/p + 1/p' = 1$ .
Convolution: Lp spaces are closed under convolution with L1 functions, with $\|f * g\|_p \leq \|f\|_1 \|g\|_p$ .

These connections make Lp spaces essential for harmonic analysis and signal processing.

Remark 9.8: Geometry of Lp Spaces

The geometry of Lp spaces varies with $p$ :

L2 (Hilbert space): Has an inner product, allowing geometric concepts like angles, orthogonality, and projections.
Lp for p ≠ 2: No inner product, but still have norm and metric structure. The unit ball shape changes with $p$ .
Uniform convexity: Lp spaces are uniformly convex for $1 < p < \infty$ , which has important consequences for optimization and fixed point theory.
Duality: The geometry of Lp is closely related to the geometry of Lq through the duality pairing.

Understanding the geometry of Lp spaces is crucial for many applications in analysis and optimization.

Remark 9.1: Key Insights

Key takeaways:

Lp spaces are complete normed vector spaces (Banach spaces)
Hölder's inequality is fundamental for many estimates
Minkowski's inequality ensures the triangle inequality holds
The dual of Lp (1 < p < ∞) is Lq where p and q are conjugate
L2 is special as a Hilbert space with geometric structure
Completeness (Riesz-Fischer) is crucial for analysis

Practice Quiz

Lp Spaces & Duality

Questions

Correct

Accuracy

For

1 \leq p < \infty

, the

L^p

norm is defined as:

Easy

Not attempted

Hölder's inequality states that if

f \in L^p

and

g \in L^q

where

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

, then:

Medium

Not attempted

Minkowski's inequality states that for

f, g \in L^p

with

1 \leq p \leq \infty

Medium

Not attempted

The Riesz-Fischer theorem states that:

Hard

Not attempted

For

1 < p < \infty

, the dual space of

L^p

is:

Hard

Not attempted

The space

L^2

is special because:

Easy

Not attempted

p < q

, then for functions on a set of finite measure:

Medium

Not attempted

The conjugate exponent of

p

q

where:

Easy

Not attempted

A linear functional

\phi

L^p

(for

1 < p < \infty

) can be represented as:

Hard

Not attempted

The

L^\infty

norm is defined as:

Medium

Not attempted

Frequently Asked Questions

What is an Lp space?

An Lp space (for 1 ≤ p ≤ ∞) consists of measurable functions f such that $\int |f|^p < \infty$ (or $\|f\|_\infty < \infty$ for p = ∞), modulo functions that are zero almost everywhere. It's a complete normed vector space (Banach space).

What is Hölder's inequality?

Hölder's inequality states that if $f \in L^p$ and $g \in L^q$ where $\frac{1}{p} + \frac{1}{q} = 1$, then $\|fg\|_1 \leq \|f\|_p \|g\|_q$. It's a generalization of the Cauchy-Schwarz inequality (which is the case p = q = 2).

What is Minkowski's inequality?

Minkowski's inequality is the triangle inequality for Lp spaces: $\|f + g\|_p \leq \|f\|_p + \|g\|_p$. It ensures that the Lp norm satisfies the triangle inequality, making Lp a normed space.

Why is L2 special?

L2 is a Hilbert space, meaning it has an inner product $\langle f, g \rangle = \int fg$ that induces the norm. This makes it particularly nice for analysis, as we can use geometric intuition and have concepts like orthogonality.

What is the dual space of Lp?

For 1 < p < ∞, the dual space (space of bounded linear functionals) of Lp is isometrically isomorphic to Lq where $\frac{1}{p} + \frac{1}{q} = 1$. This is the Riesz representation theorem. For p = 1, the dual is L∞, but the converse is more complicated.

What does it mean for Lp to be complete?

Completeness means that every Cauchy sequence in Lp converges to a function in Lp. This is the Riesz-Fischer theorem and is crucial for many applications, as it ensures that limits of Lp functions remain in Lp.

How do Lp spaces relate to each other?

On sets of finite measure, we have $L^q \subset L^p$ when p < q. On the whole real line, there's no general inclusion, but L1 ∩ L∞ is contained in all Lp. The Lp norms are related through interpolation theory.