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Spaces of Integrable Functions

Learning Goals

  1. The spaces of integrable functions.

  2. What do LpL^p norms measure? Intuition via bump functions.

  3. The Rainbow of function spaces.

Spaces of Integrable Functions

Definition 1 (LpL^p Spaces)

For 1≤p<∞1 \leq p < \infty we define the spaces of pp-integrable functions

Lp(Ω)={f:Ω→R:∥f∥p=(∫Ω∣f(x)∣p dμ)1/p<∞}/∼L^p(\Omega) = \left\{f : \Omega \to \mathbb{R} : \|f\|_p = \left( \int_{\Omega} |f(x)|^p \, d\mu \right)^{1/p} < \infty \right\} \Big/ \sim
(1)

where f∼gf \sim g if f=gf = g almost everywhere (i.e., μ({x:f(x)≠g(x)})=0\mu(\{x : f(x) \neq g(x)\}) = 0).

For p=∞p = \infty we define

L∞(Ω)={f:Ω→R:∥f∥∞=ess sup⁡x∈Ω∣f(x)∣<∞}/∼L^\infty(\Omega) = \left\{f : \Omega \to \mathbb{R} : \|f\|_\infty = \operatorname{ess\,sup}_{x \in \Omega} |f(x)| < \infty \right\} \Big/ \sim
(2)
Why equivalence classes?

Strictly speaking, elements of LpL^p are not functions but equivalence classes of functions that agree almost everywhere. This quotient is essential: without it, ∥⋅∥p\|\cdot\|_p is only a seminorm, not a norm. The problem is that ∥f∥p=0\|f\|_p = 0 does not imply f=0f = 0 pointwise—only f=0f = 0 a.e. For instance, the function that equals 1 at a single point and 0 elsewhere has LpL^p norm zero but is not the zero function. Passing to equivalence classes identifies all such functions with the zero class, making ∥f∥p=0  ⟹  f=0\|f\|_p = 0 \implies f = 0 in LpL^p.

This is a fundamental difference from CkC^k spaces, where functions are determined by their pointwise values and the sup-norm is a genuine norm without any quotienting.

What Do LpL^p Norms Measure?

Different norms capture different features of a function. To build intuition, consider a smooth bump function ϕ∈Cc∞(R)\phi \in \mathcal{C}^\infty_c(\mathbb{R}) with ϕ≥0\phi \geq 0, supp⁡(ϕ)⊂[−1,1]\operatorname{supp}(\phi) \subset [-1, 1], and ∥ϕ∥∞=1\|\phi\|_\infty = 1.

We construct two families of functions that independently control height and width.

Scaling the height. Define fA(x)=A ϕ(x)f_A(x) = A\,\phi(x) for A>0A > 0. This rescales the amplitude without changing the support. Then:

∥fA∥∞=A,∥fA∥1=A∥ϕ∥1,∥fA∥p=A∥ϕ∥p.\|f_A\|_\infty = A, \qquad \|f_A\|_1 = A\|\phi\|_1, \qquad \|f_A\|_p = A\|\phi\|_p.
(3)

All norms see the height equally — no surprise, since we are simply rescaling the function values.

Scaling the width. Define gR(x)=ϕ(x/R)g_R(x) = \phi(x/R) for R>0R > 0. This stretches the support to [−R,R][-R, R] without changing the peak value. Then:

∥gR∥∞=∥ϕ∥∞=1\|g_R\|_\infty = \|\phi\|_\infty = 1
(4)

is independent of RR — the L∞L^\infty norm is completely insensitive to how wide the function is. On the other hand, by a change of variables y=x/Ry = x/R:

∥gR∥pp=∫∣gR(x)∣p dx=R∫∣ϕ(y)∣p dy=R ∥ϕ∥pp\|g_R\|_p^p = \int |g_R(x)|^p \, dx = R \int |\phi(y)|^p \, dy = R\,\|\phi\|_p^p
(5)

so that ∥gR∥p=R1/p∥ϕ∥p\|g_R\|_p = R^{1/p}\|\phi\|_p, which grows as we widen the support.

Remark 1 (Interpretation of LpL^p Norms)

This analysis reveals a fundamental distinction:

  • L∞L^\infty measures only the height (peak value) of a function. It is completely insensitive to the width of the support.

  • L1L^1 measures the total mass — it averages the function over the domain, so both height and width contribute equally.

  • LpL^p for 1<p<∞1 < p < \infty interpolates between these extremes: larger pp gives more weight to peak values and less to the spread.

This is why L∞L^\infty convergence (uniform convergence) is a much stronger requirement than L1L^1 convergence: a sequence of tall, narrow spikes can have L1L^1 norm tending to zero while the L∞L^\infty norm remains constant.

LpL^p is a Banach Space

Theorem 1 (LpL^p is a Banach Space)

LpL^p is a Banach space.

Proof:

We must show that every Cauchy sequence in LpL^p converges. Let {fn}\{f_n\} be a Cauchy sequence in LpL^p i.e. ∀ε>0\forall \varepsilon > 0 ∃N:\exists N : ∥fn−fm∥<ε\|f_n - f_m\| < \varepsilon ∀n,m>N\forall n,m > N.

Recall that we can extract a subsequence from a Cauchy sequence (I believe this is often called a diagonal Cantor argument?) by selecting an increasing sequence njn_j such that ∥fnk+1−fnk∥<2−k, k=1,2,…\|f_{n_{k+1}} - f_{n_k}\| < 2^{-k},\ k = 1, 2, \dots

Define Gk:=∑k=1K∣fnk+1−fnk∣G_k := \sum_{k=1}^{K} |f_{n_{k+1}} - f_{n_k}|

and G=lim⁡kGkG = \lim_k G_k. By Minkowski’s inequality we have ∥Gk∥p≤∑k=1K∥fnk+1−fnk∥≤1\|G_k\|_p \leq \sum_{k=1}^{K} \|f_{n_{k+1}} - f_{n_k}\| \leq 1

for all kk. Since GkG_k is an increasing sequence we apply the Monotone convergence theorem ∫Gpdμ=∫lim⁡kGkpdμ=lim⁡k∫Gkpdμ≤1\int G^p d\mu = \int \lim_k G_k^p d\mu = \lim_k \int G_k^p d\mu \leq 1

This implies that G∈LpG \in L^p and thus G(x)<∞G(x) < \infty almost everywhere. Define f(x)=fn1+∑k=1∞fnk+1−fnkf(x) = f_{n_1} + \sum_{k=1}^{\infty} f_{n_{k+1}} - f_{n_k}

Note that this is a telescoping sum so we have that fnk→f(x)f_{n_k} \to f(x) almost everywhere. We also have ∥f∥p≤∥fn1∥p+∥G∥p<∞\|f\|_p \leq \|f_{n_1}\|_p + \|G\|_p < \infty

Thus f∈Lpf \in L^p. Finally, we apply Fatou’s lemma (to show that fn→ff_n \to f in LpL^p) ∫∥f−fn∥pdμ=∫lim⁡k→∞∥fnk−fn∥pdμ≤lim inf⁡k→∞∫∥fnk−fn∥pdμ≤εp\int \|f - f_n\|^p d\mu = \int \lim_{k \to \infty} \|f_{n_k} - f_n\|^p d \mu \leq \liminf_{k \to \infty} \int \|f_{n_k} - f_n\|^p d \mu \leq \varepsilon^p

when nk>Nn_k > N. Thus we have that ∥f−fn∥→0\|f - f_n\| \to 0.

The Rainbow of Function Spaces

Remark 2 (The Rainbow of Function Spaces)

A key result from this lecture is the “rainbow” of function spaces. The spaces of continuous and differentiable functions naturally are nested. Similarly, the spaces of integrable functions are nested by Hölder’s inequality for bounded domains.

In particular, spaces of integrable functions are only nestable when the measure of the domain is finite or the counting measure (in this case the role of pp and qq reverse). Thus crucially spaces of integrable functions with unbounded domains are not nested.

Applied Functional Analysis
Spaces of Continuous and Differentiable Functions
Applied Functional Analysis
Density and Approximation