\(L^p\) spaces turn measurable functions modulo almost-everywhere equality into Banach spaces. They are the analytic bridge between measure theory and functional analysis.
Norms and equivalence classes
Functions equal almost everywhere represent the same \(L^p\) element.
Definition. For \(1\le p<\infty\), \(L^p(\mu)\) consists of measurable functions modulo almost-everywhere equality with \[ \|f\|_p=\left(\int|f|^p\,\mathrm{d}\mu\right)^{1/p}<\infty. \] For \(p=\infty\), \[ \|f\|_\infty =\operatorname*{ess\,sup}|f| =\inf\{M\ge0:|f|\le M\text{ a.e.}\}. \]
Proposition. \(\|f\|_p=0\) if and only if \(f=0\) almost everywhere.
Proof. If \(f=0\) almost everywhere, the integral is zero. Conversely, if \(\int |f|^p=0\), then for every \(n\), the set \(\{|f|>1/n\}\) has measure zero by Markov's inequality. The union over \(n\) is \(\{|f|>0\}\), so \(f=0\) almost everywhere.
H?lder inequality
H?lder controls products by dual norms.
Theorem. If \(1\lt p\lt \infty\), \(1/p+1/q=1\), then \(\int |fg|\,\mathrm{d}\mu\le\|f\|_p\|g\|_q\).
Proof. Normalize \(\|f\|_p=\|g\|_q=1\). Young's inequality \(ab\le a^p/p+b^q/q\) gives \(|fg|\le |f|^p/p+|g|^q/q\). Integrating yields the normalized result; scaling gives the general case.
Minkowski inequality
The triangle inequality is the nontrivial part of the \(L^p\) norm.
Theorem. For \(1\le p\le\infty\), \(\|f+g\|_p\le\|f\|_p+\|g\|_p\).
Proof. The cases \(p=1,\infty\) are direct. For \(1\lt p\lt \infty\), write \(|f+g|^p\le |f||f+g|^{p-1}+|g||f+g|^{p-1}\). Apply H?lder to both terms with conjugate exponent \(q=p/(p-1)\), then divide by \(\|f+g\|_p^{p-1}\) when nonzero.
Completeness
\(L^p\) spaces are Banach spaces.
Theorem. For \(1\le p\le\infty\), \(L^p(\mu)\) is complete.
Proof. Let \((f_n)\) be Cauchy and first take \(p<\infty\). Select \(n_k\) so that \[ \|f_{n_{k+1}}-f_{n_k}\|_p\le2^{-k}. \] Put \(g_k=|f_{n_{k+1}}-f_{n_k}|\) and \(G_N=\sum_{k=1}^Ng_k\). Minkowski gives \[ \|G_N\|_p\le\sum_{k=1}^N2^{-k}\le1. \] By monotone convergence, \[ \int\left(\sum_{k=1}^{\infty}g_k\right)^p\,\mathrm{d}\mu =\lim_N\|G_N\|_p^p\le1. \] Thus \(\sum_kg_k<\infty\) almost everywhere, so \(f_{n_k}\) converges pointwise a.e. to some \(f\). For fixed \(k\), Fatou gives \[ \|f_{n_k}-f\|_p \le\liminf_{\ell\to\infty}\|f_{n_k}-f_{n_\ell}\|_p, \] which tends to zero as \(k\to\infty\). Since the original sequence is Cauchy, it converges to the same \(f\) in norm.
For \(p=\infty\), choose a subsequence with successive essential-sup differences at most \(2^{-k}\). Outside the union of the corresponding null exceptional sets, the series of pointwise differences converges uniformly. Its limit is essentially bounded, the subsequence converges in essential sup, and Cauchy convergence again restores the full sequence.
Proposition (comparison on finite measure spaces). If \(\mu(X)<\infty\) and \(1\le p<q\le\infty\), then \[ L^q(\mu)\subset L^p(\mu), \qquad \|f\|_p\le\mu(X)^{1/p-1/q}\|f\|_q. \]
Proof. For finite \(q\), apply H?lder to \(|f|^p\cdot1\) with exponents \(q/p\) and \(q/(q-p)\): \[ \int|f|^p \le\left(\int|f|^q\right)^{p/q} \mu(X)^{1-p/q}. \] Take \(p\)-th roots. The \(q=\infty\) case follows from \(|f|^p\le\|f\|_\infty^p\) almost everywhere.
Hilbert structure of \(L^2\)
\(L^2\) is special because its norm comes from an inner product.
Example. In \(L^2\), \[ \langle f,g\rangle=\int f\overline g\,\mathrm{d}\mu, \qquad \|f+g\|_2^2+\|f-g\|_2^2 =2\|f\|_2^2+2\|g\|_2^2. \] The parallelogram identity is what distinguishes Hilbert geometry from general \(L^p\) geometry. Orthogonal projection, conditional expectation, Fourier series, and least-squares estimation all exploit it.