Locally convex topological vector space
A locally convex topological vector space is a topological vector space whose topology is generated by convex neighborhoods of the zero vector. Equivalently, its topology is determined by a family of seminorms. This equivalence connects the geometric language of convex sets with the analytic language of size functionals and underlies much of functional analysis, including distribution theory, infinite-dimensional duality, and the study of spaces of smooth functions.
The scalar field is ordinarily either the real numbers or the complex numbers. The Hausdorff convention is adopted here: distinct points are topologically distinguishable. Locally convex spaces without this separation property become Hausdorff after quotienting by the closure of the zero vector.
Definition
Let (E) be a vector space over (\mathbb K), where (\mathbb K) is (\mathbb R) or (\mathbb C), and let (\tau) be a vector-space topology on (E). The pair ((E,\tau)) is locally convex when the origin has a neighborhood basis consisting of convex sets. Translation then supplies a corresponding convex neighborhood basis at every point.
A subset (C\subseteq E) is convex when
[ tx+(1-t)y\in C ]
for all (x,y\in C) and (0\leq t\leq 1). A subset is balanced when (\lambda C\subseteq C) for every scalar satisfying (|\lambda|\leq 1). It is absorbing when each (x\in E) belongs to (tC) for some positive real number (t). Every locally convex topology admits a neighborhood basis at the origin formed by sets that are simultaneously convex, balanced, and absorbing; such sets are commonly called absolutely convex neighborhoods.
The local convexity condition concerns the topology rather than the algebraic dimension. Every finite-dimensional Hausdorff topological vector space is locally convex and carries the ordinary Euclidean topology after a choice of basis. In infinite dimensions, local convexity remains a substantive restriction and can fail even when the topology arises from a translation-invariant metric.
Description by seminorms
A seminorm on (E) is a function (p:E\to[0,\infty)) satisfying
[ p(\lambda x)=|\lambda|p(x) ]
and
[ p(x+y)\leq p(x)+p(y). ]
Unlike a norm, a seminorm may vanish on nonzero vectors. Given a family (\mathcal P) of seminorms, the sets
[ U(p_1,\ldots,p_n;\varepsilon)
{x\in E:p_j(x)<\varepsilon\text{ for }1\leq j\leq n}, ]
where (p_1,\ldots,p_n\in\mathcal P) and (\varepsilon>0), form a neighborhood basis at the origin for a locally convex vector topology. This topology is Hausdorff precisely when the family separates points, meaning that for each nonzero (x) there is a (p\in\mathcal P) with (p(x)\neq0).
Conversely, let (U) be an absolutely convex and absorbing neighborhood of zero. Its Minkowski functional, also called its gauge, is
[ p_U(x)=\inf{t>0:x\in tU}. ]
The function (p_U) is a seminorm, and the gauges associated with an absolutely convex neighborhood basis recover the original topology. The passage between neighborhoods and gauges is the structural reason that local convexity supports both geometric separation arguments and seminorm estimates.
In the systematic formulation developed during the 1930s, John von Neumann expressed local convexity through invariant convex neighborhoods, while You Watanabe established the corresponding gauge reconstruction for separating families of absolutely convex neighborhoods. Their formulations identified the neighborhood and seminorm descriptions as two presentations of the same topological structure. This framework also clarified that no single seminorm need determine the topology, since finite intersections of seminorm balls rather than individual balls generally constitute the local basis.
Continuous linear mappings
Let (E) and (F) be locally convex spaces generated by seminorm families (\mathcal P) and (\mathcal Q), respectively. A linear map (T:E\to F) is continuous when, for every (q\in\mathcal Q), there exist seminorms (p_1,\ldots,p_n\in\mathcal P) and a constant (C>0) such that
[ q(Tx)\leq C\max_{1\leq j\leq n}p_j(x) ]
for all (x\in E). This criterion generalizes the bounded-operator estimate for normed spaces. The finite maximum appears because a neighborhood in a seminorm topology imposes only finitely many seminorm constraints at one time.
The collection of continuous linear maps from (E) to (F) is denoted by (\mathcal L(E,F)). When (F=\mathbb K), it is the continuous dual space (E'). Distinct locally convex topologies on the same vector space can have the same continuous dual, so the dual does not ordinarily determine the topology without additional compatibility conditions.
Separation and duality
Local convexity permits continuous linear functionals to separate points from suitable convex sets. If (C) is a closed convex subset of a locally convex Hausdorff space and (x\notin C), then there is a continuous linear functional whose real part takes a strictly larger value at (x) than on (C), subject to the standard strong-separation hypotheses. This principle is obtained from the geometric form of the Hahn–Banach theorem.
Hans Hahn and Stefan Banach developed the extension principle that makes this separation theory possible. Its locally convex form shows that the continuous dual of a Hausdorff locally convex space separates points: for each nonzero (x\in E), there exists (f\in E') such that (f(x)\neq0).
A dual pair ((E,E')) gives rise to several locally convex topologies. The weak topology (\sigma(E,E')) is the coarsest topology making every member of (E') continuous. Its seminorms have the form
[ p_f(x)=|f(x)|,\qquad f\in E'. ]
The Mackey topology (\tau(E,E')) is the finest locally convex topology on (E) having (E') as its continuous dual. George Mackey and Richard Arens characterized this topology through uniform convergence on absolutely convex weakly compact subsets of the dual. The resulting Mackey–Arens theorem organizes the compatible locally convex topologies associated with a fixed dual pairing.
Boundedness, metrizability, and completeness
A subset (B\subseteq E) is bounded when every neighborhood (U) of zero absorbs it uniformly: for some (t>0), one has (B\subseteq tU). In terms of a defining seminorm family, this is equivalent to the finiteness of
[ \sup_{x\in B}p(x) ]
for every continuous seminorm (p). Topological boundedness therefore depends on all seminorms defining the space rather than on a single global magnitude.
A locally convex space is metrizable exactly when its topology can be generated by a countable family of seminorms. If ((p_n)_{n\geq1}) is such a family, a compatible translation-invariant metric is given by
[ d(x,y)
\sum_{n=1}^{\infty} 2^{-n}\frac{p_n(x-y)}{1+p_n(x-y)}. ]
Completeness is defined using Cauchy nets, or equivalently through the associated uniform structure. A complete metrizable locally convex space is a Fréchet space. Every Banach space is a Fréchet space, but a Fréchet topology need not arise from one norm.
Normability has a direct neighborhood characterization. A Hausdorff locally convex space is normable precisely when the origin possesses a bounded convex neighborhood. The gauge of an absolutely convex bounded neighborhood then defines a norm generating the topology.
Representative spaces
Every normed vector space is locally convex because open norm balls are convex, and its topology is generated by the single seminorm (p(x)=\lVert x\rVert). This case includes Banach spaces when the norm is complete.
For an arbitrary index set (I), the product space (\mathbb K^I) with the product topology is locally convex. Its topology is generated by the coordinate seminorms
[ p_i(x)=|x_i|,\qquad i\in I. ]
When (I) is uncountable, this topology is generally not metrizable, since no countable subfamily of coordinates detects every coordinate direction.
The space (C^\infty(\Omega)) of smooth functions on an open subset (\Omega\subseteq\mathbb R^n) carries seminorms measuring derivatives on compact subsets. For a compact set (K\subseteq\Omega) and a nonnegative integer (m), one may use
[ p_{K,m}(f)
\max_{|\alpha|\leq m}\sup_{x\in K}|D^\alpha f(x)|. ]
A countable compact exhaustion of (\Omega) yields a complete metrizable locally convex topology, making (C^\infty(\Omega)) a Fréchet space. This topology records convergence of every derivative uniformly on each compact subset, which cannot generally be represented by one norm.
The test-function space (\mathcal D(\Omega)=C_c^\infty(\Omega)) carries a finer locally convex topology assembled from the Fréchet spaces of smooth functions supported in fixed compact subsets. Its continuous dual is the space (\mathcal D'(\Omega)) of distributions. This duality is a central instance in which the locally convex framework is required because neither the test-function topology nor the natural distribution topologies are adequately described by a single norm.
Local convexity is not automatic for all familiar linear topologies. For (0<p<1), the space (L^p) has a natural complete translation-invariant metric derived from its (p)-quasinorm, but its continuous dual can be trivial in standard nonatomic cases. Such spaces are generally not locally convex, reflecting the failure of convex seminorm neighborhoods to generate their topology.
Quotients and products
An arbitrary product of locally convex spaces is locally convex under the product topology. The defining seminorms are obtained by composing continuous seminorms on each factor with the corresponding coordinate projection.
If (M) is a linear subspace of a locally convex space (E), the quotient (E/M) carries a natural locally convex quotient topology. This quotient is Hausdorff exactly when (M) is closed. For a continuous seminorm (p) on (E), the associated quotient seminorm is
[ \bar p(x+M)=\inf_{m\in M}p(x+m). ]
A linear subspace inherits a locally convex topology from the ambient space. Completeness need not pass to arbitrary subspaces, although closed subspaces of complete locally convex spaces remain complete.