$L^p$ Spaces

For $1 \leq p < \infty$, define

$L^p(\mu) = \left\{f : X \to \mathbb{R} \text{ measurable} : \int_X |f|^p\, d\mu < \infty\right\}$

with the norm $\|f\|_p = \left(\int |f|^p\, d\mu\right)^{1/p}$.

For $p = \infty$, define $L^\infty(\mu) = \{f : X \to \mathbb{R} \text{ measurable} : \text{ess sup}|f| < \infty\}$ where $\|f\|_\infty = \mathrm{ess\,sup}|f|$.

Remark. Elements of $L^p$ are equivalence classes of functions equal a.e. The norm $\|\cdot\|_p$ is well-defined on equivalence classes.

Theorem 7.1 (Holder’s Inequality). Let $1 \leq p, q \leq \infty$ with $1/p + 1/q = 1$. If $f \in L^p(\mu)$ and $g \in L^q(\mu)$, then $fg \in L^1(\mu)$ and

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

Proof sketch. Use Young’s inequality: $ab \leq a^p/p + b^q/q$ for $a, b \geq 0$. Set $a = |f|/\|f\|_p$ and $b = |g|/\|g\|_q$ and integrate. $\blacksquare$

Special case ($p = q = 2$): This reduces to the Cauchy-Schwarz inequality: $\|fg\|_1 \leq \|f\|_2 \|g\|_2$.

Theorem 7.2 (Minkowski’s Inequality). For $1 \leq p \leq \infty$ and $f, g \in L^p(\mu)$:

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

Proof sketch (for $1 < p < \infty$). Write $|f + g|^p = |f + g| \cdot |f + g|^{p-1}$. Apply Holder’s inequality with conjugate exponents $p$ and $q = p/(p-1)$:

$\int |f + g|^p \leq \|f\|_p \|f + g\|_p^{p/q} + \|g\|_p \|f + g\|_p^{p/q}$

Divide both sides by $\|f + g\|_p^{p/q}$. $\blacksquare$

Theorem 7.3. $(L^p(\mu), \|\cdot\|_p)$ is a Banach space for every $1 \leq p \leq \infty$.

Proof sketch. Let $\{f_n\}$ be a Cauchy sequence in $L^p$. Extract a subsequence $f_{n_k}$ with $\|f_{n_{k+1}} - f_{n_k}\|_p < 2^{-k}$. Define $g = |f_{n_1}| + \sum_{k=1}^{\infty} |f_{n_{k+1}} - f_{n_k}|$. By the triangle inequality, $\|g\|_p \leq \|f_{n_1}\|_p + 1$. So $g \in L^p$, hence $g < \infty$ a.e., meaning $f_{n_k}$ converges a.e. to some $f$. Show $f \in L^p$ and $f_n \to f$ in $L^p$-norm. $\blacksquare$

Theorem 7.4. $L^2(\mu)$ is a Hilbert space with inner product $\langle f, g \rangle = \int f\overline{g}\, d\mu$.

Proposition 7.5. If $\mu$ is a finite measure and $1 \leq p < q \leq \infty$, then $L^q(\mu) \subseteq L^p(\mu)$. In particular, $L^\infty(\mu) \subseteq L^2(\mu) \subseteq L^1(\mu)$.

Proof. Apply Holder’s inequality with $q/p$ and its conjugate. $\blacksquare$

Theorem 7.6 (Riesz Representation for $L^p$). For $1 < p < \infty$, the dual space $(L^p)^*$ is isometrically isomorphic to $L^q$, where $1/p + 1/q = 1$. The pairing is:

$\langle f, g \rangle = \int f g\, d\mu, \quad f \in L^p, g \in L^q$

This also holds for $p = 1$ provided $\mu$ is $\sigma$-finite (the dual of $L^1$ is $L^\infty$). For $p = \infty$, the dual is strictly larger than $L^1$ (except for finite-dimensional spaces).

Proposition 7.7. For $1 < p < \infty$, $L^p$ spaces are uniformly convex: for any $\varepsilon > 0$, there exists $\delta > 0$ such that if $\|f\|_p = \|g\|_p = 1$ and $\|f - g\|_p \geq \varepsilon$, then $\|(f + g)/2\|_p \leq 1 - \delta$.

Uniform convexity implies reflexivity for $1 < p < \infty$ and guarantees the existence and uniqueness of best approximations in closed convex subspaces.

Proposition 7.8 (Density of Simple Functions). Simple functions are dense in $L^p(\mu)$ for $1 \leq p < \infty$. For the Lebesgue measure on $\mathbb{R}^n$, the following are also dense:

1. Step functions (finite linear combinations of characteristic functions of rectangles)
2. Continuous functions with compact support $C_c(\mathbb{R}^n)$
3. Smooth functions with compact support $C_c^\infty(\mathbb{R}^n)$

Proposition 7.9. If $f_n \to f$ in $L^p$, then there exists a subsequence $f_{n_k}$ that converges pointwise a.e. to $f$. The converse is false: pointwise convergence a.e. does not imply $L^p$ convergence (counterexample: $f_n = n\chi_{[0, 1/n]}$ on $[0, 1]$ converges pointwise to $0$ but $\|f_n\|_1 = 1$ for all $n$).

Theorem 7.10 (Dominated Convergence in $L^p$). If $f_n \to f$ a.e. and there exists $g \in L^p$ such that $|f_n| \leq g$ a.e. for all $n$, then $f_n \to f$ in $L^p$.

Problem. Let $f(x) = x^{-1/2}$ on $(0, 1)$ with Lebesgue measure. For which $p$ does $f \in L^p(0, 1)$?

Problem. Show that if $f \in L^p \cap L^q$ with $1 \leq p < q \leq \infty$, then $f \in L^r$ for all $r \in [p, q]$.

Definition. The weak $L^p$ space $L^{p,\infty}(\mu)$ consists of measurable functions for which

$[f]_{p,\infty} = \sup_{t > 0} t\, \mu(\{x : |f(x)| > t\})^{1/p} < \infty$

Weak $L^p$ spaces are larger than $L^p$: $L^p \subseteq L^{p,\infty}$ with $\|f\|_{p,\infty} \leq \|f\|_p$.

Example. The function $f(x) = 1/|x|^{1/p}$ on $\mathbb{R}$ is in $L^{p,\infty}$ but not in $L^p$ (the singularity is just barely non-integrable in the $L^p$ sense).

Lorentz spaces $L^{p,q}(\mu)$ refine the $L^p$ scale, with $L^{p,p} = L^p$ and $L^{p,\infty}$ being weak $L^p$. They are important in interpolation theory and harmonic analysis.

Problem. Show that if $\mu(X) < \infty$, then $\lim_{p \to \infty} \|f\|_p = \|f\|_\infty$ for $f \in L^\infty(\mu)$.