Lp Spaces - Main Theorem

TheoremRiesz-Fischer Theorem

For $1 \leq p \leq \infty$ , the space $L^p(X, \mathcal{F}, \mu)$ is a Banach space: it is complete under the $L^p$ norm.

Explicitly, if $\{f_n\}$ is a Cauchy sequence in $L^p$ , meaning that for every $\epsilon > 0$ there exists $N$ such that: $\|f_n - f_m\|_p < \epsilon \text{ for all } n, m \geq N$

then there exists $f \in L^p$ such that $\|f_n - f\|_p \to 0$ .

Moreover, there exists a subsequence $\{f_{n_k}\}$ that converges to $f$ almost everywhere.

The Riesz-Fischer Theorem is fundamental because completeness allows us to take limits freely in $L^p$ , making these spaces suitable for solving equations and optimization problems. It guarantees that $L^p$ has no "holes" - every Cauchy sequence converges within the space.

ExampleApplication to Fourier Series

Consider the trigonometric system $\{e^{inx}\}_{n \in \mathbb{Z}}$ in $L^2([-\pi, \pi])$ . These functions form an orthonormal basis.

For any $f \in L^2[-\pi, \pi]$ , the Fourier series is: $f \sim \sum_{n=-\infty}^{\infty} c_n e^{inx} \text{ where } c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx$

The partial sums $S_N = \sum_{|n| \leq N} c_n e^{inx}$ form a Cauchy sequence in $L^2$ (by Bessel's inequality). By Riesz-Fischer, this sequence converges in $L^2$ norm to some function $g \in L^2$ .

Furthermore, $g = f$ a.e., establishing that the Fourier series converges to $f$ in $L^2$ . Without completeness, we couldn't guarantee that the limit is in $L^2$ .

Remark

Important consequences of the Riesz-Fischer Theorem:

$L^2$ is a Hilbert space: Combined with the inner product $\langle f, g \rangle = \int fg \, d\mu$ , $L^2$ becomes a complete inner product space, which is the definition of a Hilbert space.
Dual spaces: The completeness of $L^p$ enables the identification of dual spaces. For $1 < p < \infty$ , the dual of $L^p$ is isometrically isomorphic to $L^q$ where $1/p + 1/q = 1$ .
Weak convergence: The theorem implies that bounded sequences in $L^p$ have weakly convergent subsequences (by the Banach-Alaoglu theorem).
Fixed point theorems: Completeness allows application of contraction mapping principles and other fixed point theorems in $L^p$ spaces.

The proof uses the fact that a Cauchy sequence has a subsequence converging a.e., and then Fatou's Lemma to show the limit is in $L^p$ with the correct norm.