List of Function Spaces

Introduction

This article lists the function spaces the corpus meets. A function space is a collection of functions carrying a topology — usually a norm or a family of seminorms — in which limits may be taken, and the rows below name each such space, record the norm or the topology that makes it a space of analysis, record whether that topology is complete, and point to the article that introduces the space. Every row points to an article; this article introduces nothing and proves nothing. The spaces are grouped by the layer that introduces them — the measurable and sequence spaces, the continuous and smooth spaces, the smoothness scales, and the analytic and test-function spaces — and the spaces that fail completeness, or that carry no norm at all, are recorded beside the complete ones as non-examples, so that the class is visibly a proper part of the class above it.

The Measurable and Sequence Spaces

Object	The norm it carries, and its completeness	Introduced in
$L^p(\mu)$, $1 \leq p < \infty$	$\lVert f\rVert_p = \left(\int \lvert f\rvert^p d\mu\right)^{1/p}$; complete by the Riesz–Fischer theorem	Measure Theory and Integration
$L^\infty(\mu)$	essential supremum norm $\lVert f\rVert_\infty = \inf\{c : \lvert f\rvert \leq c \text{ a.e.}\}$; complete	Measure Theory and Integration
$L^2(\mu)$	the norm of the inner product $\langle f,g\rangle = \int f\bar g\,d\mu$; complete, and the basic Hilbert space	Banach and Hilbert Spaces
$\ell^p$, $1 \leq p < \infty$	$\lVert(x_n)\rVert_p = \left(\sum_n \lvert x_n\rvert^p\right)^{1/p}$; complete	Normed and Banach Spaces
$\ell^\infty$	$\lVert(x_n)\rVert_\infty = \sup_n \lvert x_n\rvert$; complete	Normed and Banach Spaces
$c_0$, $c$	sup norm; complete, and closed subspaces of $\ell^\infty$	Normed and Banach Spaces
$B(X,Y)$, $C_b(X,Y)$	operator norm and sup norm; complete when $Y$ is complete	Normed and Banach Spaces; Metric, Uniform and Complete Spaces
$L^p(0,T;V)$, Bochner space	$\left(\int_0^T \lVert u(t)\rVert_V^p dt\right)^{1/p}$; complete when $V$ is Banach	Sobolev Spaces and Weak Solutions

The measurable spaces are the ones on which the integral is available, and their completeness is the completeness that makes the convergence theorems of the integral into existence theorems for limits.

The Continuous and Smooth Spaces

Object	The norm or topology it carries, and its completeness	Introduced in
$C(K)$, $K$ compact	supremum norm $\lVert f\rVert_\infty$; complete	Normed and Banach Spaces
$C_0(X)$, $C_c(X)$	sup norm; $C_0$ complete, $C_c$ dense in $C_0$	Fourier Analysis on Euclidean Spaces; Modes of Convergence
$C^k(U,Y)$, $0 \leq k < \infty$	the seminorms of the derivatives up to order $k$ on compacta; complete	Differential Calculus on Normed Spaces
$C^\infty(U) = \mathcal E(\Omega)$	the increasing seminorms $\sup_K \lvert\partial^\alpha f\rvert$, $K$ compact, $\lvert\alpha\rvert \leq n$; a Fréchet space, hence complete	Fréchet Spaces
$\mathcal O(\Omega)$, holomorphic functions	the seminorms of the supremum on compacta; a Fréchet space, hence complete and Montel	Fréchet Spaces
$B M O$, bounded mean oscillation	$\sup_Q \frac{1}{\lvert Q\rvert}\int_Q \lvert f - f_Q\rvert$, a seminorm modulo constants; complete	Besov and Triebel–Lizorkin Spaces

The Smoothness Scales

Object	The norm it carries, and its completeness	Introduced in
$W^{k,p}(\Omega)$, $W^{k,p}_0(\Omega)$	$\lVert f\rVert_{W^{k,p}}^p = \sum_{\lvert\gamma\rvert\leq k}\lVert\partial^\gamma f\rVert_p^p$; a Banach space, hence complete	Sobolev Spaces and Weak Solutions
$H^k(\Omega) = W^{k,2}(\Omega)$	the inner product $\sum_{\lvert\gamma\rvert\leq k}\langle\partial^\gamma f,\partial^\gamma g\rangle$; a Hilbert space	Sobolev Spaces and Weak Solutions
$H^{-1}(\Omega)$	the operator norm in the dual of $H^1_0(\Omega)$; complete	Sobolev Spaces and Weak Solutions
$H^s(\mathbb R^n)$, $s \in \mathbb R$	$\lVert f\rVert_{H^s}^2 = \int(1+\lvert\xi\rvert^2)^s\lvert\hat f(\xi)\rvert^2 d\xi$; complete	Fourier Analysis on Euclidean Spaces
$H^s_p$, Bessel-potential space	$\lVert f\rVert = \lVert(1-\Delta)^{s/2}f\rVert_p$; complete	Interpolation Theory
$C^{k,\alpha}$, Hölder space	$\lVert f\rVert = \sum_{\lvert\gamma\rvert\leq k}\lVert\partial^\gamma f\rVert_\infty + \sup_{x\neq y}\frac{\lvert\partial^\gamma f(x)-\partial^\gamma f(y)\rvert}{\lvert x-y\rvert^\alpha}$; complete	Besov and Triebel–Lizorkin Spaces
$\Lambda^s = C^s$, Hölder–Zygmund space	the Besov norm $B^s_{\infty,\infty}$; complete	Besov and Triebel–Lizorkin Spaces
$B^s_{p,q}$, Besov space	the mixed $\ell^q(L^p)$ norm of the Littlewood–Paley blocks; a quasi-Banach space, complete	Besov and Triebel–Lizorkin Spaces
$F^s_{p,q}$, Triebel–Lizorkin space	the mixed $L^p(\ell^q)$ norm of the Littlewood–Paley blocks; complete	Besov and Triebel–Lizorkin Spaces
$H^1$, real Hardy space	$\lVert f\rVert_{H^1} = \lVert \sup_t \lvert f * \phi_t\rvert\rVert_1$; complete, and the borderline case $F^0_{1,2}$	Besov and Triebel–Lizorkin Spaces
$L^{p,q}$, Lorentz space	the norm of the decreasing rearrangement, $\lVert f\rVert = \left(\int_0^\infty (t^{1/p}f^*(t))^q \frac{dt}{t}\right)^{1/q}$; complete	Interpolation Theory

The Analytic and Test-Function Spaces

Object	The norm or topology it carries, and its completeness	Introduced in
$H^p(\mathbb D)$, Hardy space	$\lVert f\rVert_{H^p} = \sup_{0	Complex Harmonic Analysis
$A^p(\mathbb D)$, Bergman space	$\left(\frac{1}{\pi}\int_{\mathbb D}\lvert f\rvert^p dA\right)^{1/p}$; complete	Complex Harmonic Analysis
$\mathcal B$, Bloch space	$\sup_z (1-\lvert z\rvert^2)\lvert f'(z)\rvert$; complete modulo constants	Complex Harmonic Analysis
$PW_B$, Paley–Wiener space	the $L^2$ norm of the boundary function; a closed subspace of $L^2$	Complex Harmonic Analysis
$\mathcal S(\mathbb R^n)$, Schwartz class	the seminorms $p_{\alpha,\beta}(f) = \sup_x \lvert x^\alpha\partial^\beta f\rvert$; a Fréchet space, hence complete	Fourier Analysis on Euclidean Spaces; Fréchet Spaces
$\mathcal D(\Omega) = C_c^\infty(\Omega)$	the LF inductive limit of the $\mathcal D_K$; complete, and not normable	Distributions and Fundamental Solutions
$\mathcal D_K(\Omega)$	the seminorms $\sup_K \lvert\partial^\alpha f\rvert$; a Fréchet space	Distributions and Fundamental Solutions
$\mathcal D'(\Omega)$, $\mathcal E'(\Omega)$, $\mathcal S'(\mathbb R^n)$	the strong dual topologies; complete	Distributions and Fundamental Solutions

The test-function spaces are the ones whose duals are the distributions, and the last three rows are recorded here as spaces, their distribution-theoretic content belonging to the catalogue of distributions.

Spaces Without a Norm, and Spaces That Are Not Complete

Object	The property that fails	Introduced in
$L^p(\mu)$ with $0 < p < 1$	carries a quasi-norm but no norm, and is not locally convex	Locally Convex Spaces
$\mathcal D(\Omega)$, $C_c^\infty(\Omega)$	complete in its LF topology, but no norm defines that topology	Distributions and Fundamental Solutions
$\mathcal P([0,1])$, the polynomials	dense in $C[0,1]$ under the supremum norm, hence normed but not complete; its completion is $C[0,1]$	Metric, Uniform and Complete Spaces
$c_{00}$, finitely supported sequences	dense in $\ell^p$, hence normed but not complete	Normed and Banach Spaces
$C([0,1])$ under the $L^2$ norm	the norm is genuine, but the space is not complete; its completion is $L^2([0,1])$	Measure Theory and Integration
Riemann integrable functions under the supremum norm	a pointwise limit of Riemann integrable functions need not be Riemann integrable, so the space is not complete	Real Integration
a countable direct sum $\bigoplus_n X_n$ of nonzero Banach spaces	incomplete under each of the sum, maximum and Euclidean norms; its completions are the sequences with $\sum_i \lVert x_i\rVert < \infty$ and those with $\lVert x_i\rVert \to 0$	Normed and Banach Spaces

Summary

This list gathers the function spaces of the corpus in four families: the measurable spaces $L^p$ and their discrete counterparts $\ell^p$; the continuous and smooth spaces $C^k$, $C^\infty$ and the holomorphic spaces; the smoothness scales $W^{k,p}$, $H^s$, the Hölder, Besov, Triebel–Lizorkin and Hardy spaces; and the analytic and test-function spaces $H^p$, $A^p$, the Schwartz class and $\mathcal D$. Each row records the norm or the topology the space carries and whether it is complete, and the final table records the spaces for which completeness or normability fails.

Summary of Notation

The objects of this list are named rather than denoted; the few symbols used in the tables are collected here.

Symbol	Meaning
$\mathbb K$	$\mathbb R$ or $\mathbb C$
$\mu$	a measure, so that $L^p(\mu)$ is a Lebesgue space
$\Omega$, $U$	an open subset of $\mathbb R^n$, or of a normed space
$K$	a compact set
$p,q$	conjugate exponents, $1/p + 1/q = 1$
$k,\alpha,s$	differentiability order, Hölder exponent, real smoothness index
$\mathcal D$, $\mathcal S$, $\mathcal E$	test functions, Schwartz class, smooth functions
$\mathcal D'$, $\mathcal S'$, $\mathcal E'$	their duals, the spaces of distributions