List of Norms and Seminorms

Introduction

This article lists the norms, seminorms and families of seminorms that the corpus uses, together with the completeness each one confers on the space it defines. A norm is a real-valued function on a vector space that is positive definite, homogeneous and satisfies the triangle inequality; a seminorm drops positive definiteness; a family of seminorms defines a locally convex topology without necessarily defining a norm. Each row names one such object, records the space or the class of spaces it acts on and the completeness it produces there, and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The objects are grouped by the layer that introduces them, and the functions that look like norms and are not — seminorms that vanish on nonzero vectors, quasi-norms that fail the triangle inequality, norms that are not submultiplicative — are recorded beside them as non-examples.

Norms on Linear Spaces

Object	The space it defines, and the completeness it gives	Introduced in
a norm $\lVert\cdot\rVert$	a normed space; complete exactly when the space is Banach	Normed and Banach Spaces
the operator norm $\lVert T\rVert = \sup_{\lVert x\rVert\leq1}\lVert Tx\rVert$	$B(X,Y)$; complete when $Y$ is complete	Normed and Banach Spaces
the inner-product norm $\lVert x\rVert = \langle x,x\rangle^{1/2}$	an inner product space; complete exactly when it is a Hilbert space	Banach and Hilbert Spaces
the dual norm on $X^*$	the dual space $X^* = B(X,\mathbb K)$; always complete	Normed and Banach Spaces
the quotient norm	$X/M$ for a closed subspace $M$; complete when $X$ is	Normed and Banach Spaces
equivalence of norms	finitely many norms on a finite-dimensional space are equivalent, so all give the same completeness	Normed and Banach Spaces

Norms on Sequence and Function Spaces

Object	The space it defines, and the completeness it gives	Introduced in
the $\ell^p$ norm $\lVert(x_n)\rVert_p$	$\ell^p$; complete for $1 \leq p \leq \infty$	Normed and Banach Spaces
the supremum norm $\lVert f\rVert_\infty = \sup_x \lvert f(x)\rvert$	$C(K)$, $C_b(X)$, $B(X,Y)$; complete when the target is complete	Normed and Banach Spaces; Metric, Uniform and Complete Spaces
the essential supremum norm	$L^\infty(\mu)$; complete	Measure Theory and Integration
the $L^p$ norm $\lVert f\rVert_p$	$L^p(\mu)$; complete for $1 \leq p \leq \infty$	Measure Theory and Integration
the Sobolev norm $\lVert f\rVert_{W^{k,p}}$	$W^{k,p}(\Omega)$; complete	Sobolev Spaces and Weak Solutions
the homogeneous Sobolev norm, and $p^* = np/(n-p)$	the Sobolev conjugate exponent, with the embedding $W^{1,p}\hookrightarrow L^{p^*}$	Sobolev Spaces and Weak Solutions
the Bessel-potential norm $\lVert(1-\Delta)^{s/2}f\rVert_p$	$H^s_p$; complete	Interpolation Theory
the Hölder norm	$C^{k,\alpha}$; complete	Besov and Triebel–Lizorkin Spaces
the Besov norm (an $\ell^q(L^p)$ norm of the Littlewood–Paley blocks)	$B^s_{p,q}$; a quasi-norm for $p<1$ or $q<1$, complete	Besov and Triebel–Lizorkin Spaces
the Triebel–Lizorkin norm (an $L^p(\ell^q)$ norm of the blocks)	$F^s_{p,q}$; complete	Besov and Triebel–Lizorkin Spaces
the Lorentz norm	$L^{p,q}$; complete, and the interpolation norm between two $L^p$ norms	Interpolation Theory
the Hardy-space norm $\lVert f\rVert_{H^p}$	$H^p(\mathbb D)$ for $1 \leq p \leq \infty$; complete for $1 \leq p < \infty$, and $H^\infty$ is the bounded holomorphic functions	Complex Harmonic Analysis
the Paley–Wiener norm inherited from $L^2$	$PW_B$; a closed subspace of $L^2$, hence complete	Complex Harmonic Analysis
the total-variation norm $\lvert\nu\rvert(X)$	the space of signed and complex measures; complete	Measure Theory and Integration

Norms on Algebras and in Duality

Object	The space it defines, and the completeness it gives	Introduced in
the submultiplicative norm, $\lVert xy\rVert \leq \lVert x\rVert\lVert y\rVert$	a normed algebra; complete exactly when it is a Banach algebra	Topological Algebras and Banach Algebras
the Neumann-series bound $(1-x)^{-1} = \sum_n x^n$ for $\lVert x\rVert<1$	the openness of the unit group in a normed algebra	Topological Algebras and Banach Algebras
the spectral radius $r(a) = \lim_n \lVert a^n\rVert^{1/n}$	a quantity bounded above by the norm, $r(a) \leq \lVert a\rVert$, but not a norm itself	Topological Algebras and Banach Algebras
the Hilbert–Schmidt norm	$L^2(H)$; complete, an inner-product norm	Operator Algebras
the trace-class norm $\lVert T\rVert_1 = \operatorname{Tr}\lvert T\rvert$	$L^1(H)$; complete	Operator Algebras
the strong and weak operator seminorms	the initial topologies of the operator families; not norms	Operator Algebras
the polar and the Minkowski gauge $p_A$	a seminorm when $A$ is convex, balanced and absorbing; gives local convexity	Locally Convex Spaces
the seminorm of a topological vector space	a locally convex topology, complete exactly when the space is Fréchet or otherwise as stated	Locally Convex Spaces

Seminorms and Families of Seminorms

Object	The topology or property it defines	Introduced in
a seminorm $p$	a convex, balanced, absorbing unit ball $B_p = \{x : p(x)\leq1\}$; the topology of a locally convex space	Locally Convex Spaces
a generating family $\mathcal P$ of seminorms	the locally convex topology it generates; complete exactly when the resulting space is complete	Locally Convex Spaces
a basic convex neighbourhood $U_{p_1,\dots,p_n;\varepsilon}$	the neighbourhood filter of $0$	Locally Convex Spaces
an increasing sequence $(p_n)$ of seminorms	the Fréchet topology it generates	Fréchet Spaces
the $F$-norm $\lvert x\rvert = \sum_n 2^{-n}\min(1,p_n(x))$	the invariant metric $d(x,y) = \lvert x-y\rvert$; an $F$-space when complete	Fréchet Spaces
the Schwartz seminorms $p_{\alpha,\beta}(f) = \sup_x \lvert x^\alpha\partial^\beta f\rvert$	the Fréchet topology of $\mathcal S(\mathbb R^n)$; complete	Fréchet Spaces; Fourier Analysis on Euclidean Spaces
the seminorms of the derivatives on compacta	the Fréchet topology of $C^\infty(U)$ and of $\mathcal O(\Omega)$; complete	Fréchet Spaces
the seminorms $\sup_K \lvert\partial^\alpha f\rvert$ on $\mathcal D_K$	the inductive-limit topology of $\mathcal D(\Omega)$; complete and not normable	Distributions and Fundamental Solutions
the $p$-adic absolute value $\lvert\cdot\rvert_p$	an absolute value satisfying the strong triangle inequality; gives a non-Archimedean norm	Topological Modules and Vector Spaces; p-adic Integration
a valuation $v = v_p$	a non-Archimedean valuation, $\lvert x\rvert_p = p^{-v(x)}$	Topological Modules and Vector Spaces

Functions That Are Not Norms

Object	The property that fails	Introduced in
$\lvert f(0)\rvert$ on the continuous functions	vanishes on the nonzero functions that are $0$ at $0$: a seminorm and not a norm	Locally Convex Spaces
$L^p(\mu)$ with $0 < p < 1$	the triangle inequality fails; the function is a quasi-norm and not a norm, and the topology is not locally convex	Locally Convex Spaces
the constant function $0$	vanishes on every vector: the degenerate seminorm	Locally Convex Spaces
the sup norm on $C^k(U)$	does not control the derivatives, so it does not make $C^k$ complete: a norm, but not the one the class carries	Differential Calculus on Normed Spaces
the spectral radius $r(a)$	subadditive and submultiplicative, but not positive definite: a quasinilpotent element satisfies $r(a) = 0$ with $a \neq 0$	Topological Algebras and Banach Algebras

Summary

This list gathers the norms of the corpus — the operator, inner-product, supremum, $L^p$, Sobolev, Hölder, Besov, Triebel–Lizorkin and Lorentz norms, with the completeness each confers — and the seminorms, either single or in generating families, that define the locally convex, Fréchet and inductive-limit topologies. The final table records the functions that are named or used as norms and fail to be one.

Summary of Notation

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

Symbol	Meaning
$\lVert\cdot\rVert$	a norm; $\lVert T\rVert$ the operator norm
$p$	a seminorm, and $B_p$ its unit ball
$\mathcal P$	a generating family of seminorms
$\lvert\cdot\rvert_p$, $v_p$	the $p$-adic absolute value and valuation
$F$-norm, Fréchet	the invariant metric norm and the complete metrisable locally convex case
quasi-norm	a function failing the triangle inequality, as in $L^p$ with $p<1$