In a normed space $(X, \|\cdot\|)$, the topology is determined by a single quantity — the norm — and convergence means $\|x_n - x\| \to 0$. This is the setting for Banach space theory: the [linear operators page](/page/Linear%20Operators%20on%20Banach%20Spaces) and the [adjoint page](/page/The%20Adjoint%20of%20an%20Operator) develop their theory entirely within this framework. But many of the most important spaces in analysis are not normed. The space of [test functions](/page/Test%20Function) $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ carries a topology defined by convergence of all [derivatives](/page/Derivative) on compact [sets](/page/Set) — a notion that cannot be captured by any single norm. The [Schwartz space](/page/Schwartz%20Space) $\mathcal{S}(\mathbb{R}^n)$ is topologised by a countable family of seminorms measuring decay and regularity simultaneously. The [weak* topology](/page/Weak*%20Topology) on a dual space, which is the correct topology for convergence of [distributions](/page/Distribution), is defined by evaluation against elements of the predual — again not a norm topology when the predual is infinite-dimensional.

A topological vector space (TVS) is the minimal structure that makes both the vector space operations and the notion of convergence compatible. It is a vector space equipped with a topology in which addition and scalar multiplication are continuous. This generality accommodates all the examples above within a single framework. The more refined notion of a *locally convex* TVS — one whose topology is generated by seminorms — is the correct setting for duality theory: continuous linear functionals separate points, the Hahn–Banach theorem extends, and the dual space is rich enough to support a meaningful theory of [weak convergence](/page/Weak%20Convergence).

This page develops the definitions and foundational properties of [topological](/page/Topology) vector spaces, with emphasis on the locally convex case and the role of seminorms. The main examples — Fréchet spaces, the test function space $\mathcal{D}(\Omega)$, and initial topologies on dual spaces — are treated in detail.

## Motivation

[motivation] ### The Inadequacy of a Single Norm Consider the sequence of smooth [functions](/page/Function) $\varphi_n \in C_c^\infty(\mathbb{R})$ defined by $\varphi_n(x) = \eta(x - n)$, where $\eta$ is a fixed bump function supported in $[-1, 1]$. In any $L^p$ norm ($1 \le p \le \infty$), we have $\|\varphi_n\|_{L^p} = \|\eta\|_{L^p}$ — the sequence does not converge to zero. In the sup norm, $\|\varphi_n\|_\infty = \|\eta\|_\infty$. Yet in distribution theory, $\varphi_n \to 0$ in a perfectly natural sense: the supports escape to infinity, so $\int \varphi_n \psi \, d\mathcal{L}^1 \to 0$ for every test function $\psi$ with compact support. More fundamentally, the topology on $\mathcal{D}(\Omega)$ requires simultaneous control of *all derivatives on every compact subset* — an infinite family of constraints that no single norm can capture. A sequence $\varphi_n \to \varphi$ in $\mathcal{D}(\Omega)$ means: the supports are eventually contained in a common compact set, and $\sup_{x \in K} |D^\alpha \varphi_n(x) - D^\alpha \varphi(x)| \to 0$ for every multi-index $\alpha$ and every compact $K \subset \Omega$. This is a convergence notion defined by a *family* of seminorms, not by a single norm. ### Seminorms as the Replacement for Norms A seminorm $p$ on a vector space $X$ satisfies the same axioms as a norm — $p(\lambda x) = |\lambda| p(x)$ and $p(x + y) \le p(x) + p(y)$ — except that $p(x) = 0$ does not force $x = 0$. A family of seminorms $\{p_\alpha\}_{\alpha \in A}$ on $X$ defines a topology by declaring a set $U$ open if for every $x \in U$ there exist finitely many indices $\alpha_1, \dots, \alpha_n$ and $\varepsilon > 0$ such that \begin{align*} \{y \in X : p_{\alpha_k}(y - x) < \varepsilon \text{ for } k = 1, \dots, n\} \subseteq U. \end{align*} In this topology, a net $x_i \to x$ if and only if $p_\alpha(x_i - x) \to 0$ for every $\alpha \in A$. When $A$ is a singleton (a single norm), this recovers the norm topology. When $A$ is countable, the topology is metrisable (by a translation-invariant metric). When $A$ is uncountable, the topology is generally non-metrisable — this is the situation for $\mathcal{D}(\Omega)$ and for weak* topologies on non-[separable](/page/Separable) duals. ### What the Framework Must Provide The purpose of the TVS framework is to axiomatise the compatibility between the vector space structure and the topology. We need: (a) addition $(x, y) \mapsto x + y$ is continuous (so [limits](/page/Limit) of sums are sums of limits); (b) scalar multiplication $(\lambda, x) \mapsto \lambda x$ is continuous (so limits commute with scaling); and (c) the topology is Hausdorff (so limits are unique). These three requirements are enough to develop a meaningful theory of convergence, [continuity](/page/Continuity) of [linear maps](/page/Linear%20Map), and — when the topology is locally convex — duality. [/motivation]

## Definition

The definition of a topological vector space encodes the bare minimum needed for vector space operations to interact continuously with the topology. No metric, no norm, no seminorms are assumed — only a topology satisfying two compatibility conditions.

[definition:Topological Vector Space] A **topological vector space** (TVS) over $\mathbb{R}$ (or $\mathbb{C}$) is a vector space $X$ equipped with a topology $\tau$ such that the following maps are continuous: (i) **Addition:** the map $(x, y) \mapsto x + y$ from $X \times X$ (with the product topology) to $X$. (ii) **Scalar multiplication:** the map $(\lambda, x) \mapsto \lambda x$ from $\mathbb{R} \times X$ (or $\mathbb{C} \times X$) to $X$. A TVS is **Hausdorff** if distinct points have disjoint neighbourhoods. [/definition]

The continuity of addition and scalar multiplication has immediate structural consequences. Every translation $x \mapsto x + a$ is a homeomorphism (with inverse $x \mapsto x - a$), so the topology is completely determined by the neighbourhood filter at the origin: $U$ is a neighbourhood of $a$ if and only if $U - a$ is a neighbourhood of $0$. This is the TVS analogue of the fact that a norm topology is determined by the open balls centred at the origin.

Similarly, every scalar multiplication $x \mapsto \lambda x$ (with $\lambda \neq 0$) is a homeomorphism, so the topology is "scale-invariant" in the sense that $U$ is a neighbourhood of $0$ if and only if $\lambda U$ is, for every $\lambda \neq 0$.

A neighbourhood $V$ of the origin is called **balanced** if $\lambda V \subseteq V$ for every scalar $\lambda$ with $|\lambda| \le 1$, and **absorbing** if for every $x \in X$ there exists $t > 0$ with $x \in tV$. In any TVS, every neighbourhood of the origin contains a balanced neighbourhood (this follows from the continuity of scalar multiplication at $(0, 0)$), and every neighbourhood of the origin is absorbing (because $\lambda x \to 0$ as $\lambda \to 0$, so eventually $\lambda x \in V$, i.e. $x \in \lambda^{-1}V$).

[remark:Hausdorff Separation] Not every TVS is Hausdorff. The indiscrete topology $\tau = \{\varnothing, X\}$ makes any vector space into a TVS, but the only Hausdorff case is $X = \{0\}$. In practice, one always assumes the Hausdorff property. A TVS is Hausdorff if and only if $\{0\}$ is a [closed set](/page/Closed%20Set), which is equivalent to the condition that the intersection of all neighbourhoods of the origin is $\{0\}$: $\bigcap\{U : U \text{ is a neighbourhood of } 0\} = \{0\}$. For a seminorm-generated topology, this means the family of seminorms is **separating**: $p_\alpha(x) = 0$ for all $\alpha$ implies $x = 0$. [/remark]

## Locally Convex Spaces

The TVS axioms alone are too weak for a satisfactory duality theory. For instance, there exist (pathological) TVS in which the only continuous linear functional is the zero functional — the dual space is trivial. The key additional hypothesis that makes duality work is **local convexity**: the topology has a base of convex neighbourhoods at the origin. This is exactly the condition needed for the Hahn–Banach theorem to supply enough continuous linear functionals to separate points.

The connection between local convexity and seminorms is the central structural fact of the theory: a topology is locally convex if and only if it can be generated by a family of seminorms. This gives a concrete, computational handle on abstract locally convex topologies.

[definition:Seminorm] A **seminorm** on a vector space $X$ is a function $p: X \to [0, \infty)$ satisfying: (i) **Absolute homogeneity:** $p(\lambda x) = |\lambda| \, p(x)$ for all $\lambda \in \mathbb{R}$ (or $\mathbb{C}$) and $x \in X$. (ii) **Triangle inequality:** $p(x + y) \le p(x) + p(y)$ for all $x, y \in X$. A seminorm is a **norm** if additionally $p(x) = 0$ implies $x = 0$. [/definition]

[definition:Locally Convex Topological Vector Space] A **locally convex topological vector space** (locally convex TVS, or LCS) is a TVS whose topology is generated by a family of seminorms $\mathcal{P} = \{p_\alpha\}_{\alpha \in A}$. Explicitly, a base of neighbourhoods at the origin is given by the sets \begin{align*} U_{\alpha_1, \dots, \alpha_n; \varepsilon} := \{x \in X : p_{\alpha_k}(x) < \varepsilon \text{ for } k = 1, \dots, n\} \end{align*} for all finite subsets $\{\alpha_1, \dots, \alpha_n\} \subseteq A$ and $\varepsilon > 0$. The space is Hausdorff if and only if $\mathcal{P}$ is **separating**: $p_\alpha(x) = 0$ for all $\alpha \in A$ implies $x = 0$. The topology is denoted $\tau_\mathcal{P}$ and called the **initial topology** generated by $\mathcal{P}$. [/definition]

The terminology "locally convex" is justified: each basic neighbourhood $U_{\alpha_1, \dots, \alpha_n; \varepsilon}$ is convex (as an intersection of sublevel sets of seminorms, which are convex). Conversely, if a TVS has a base of convex balanced neighbourhoods at the origin, one can recover a generating family of seminorms by taking the Minkowski gauge (Minkowski functional) of each such neighbourhood: for a convex balanced absorbing set $C$, the gauge $p_C(x) := \inf\{t > 0 : x \in tC\}$ is a seminorm, and the sublevel set $\{p_C < 1\}$ is the interior of $C$.

### Why Local Convexity Is Needed

The Hahn–Banach extension theorem — the foundation of all duality theory — requires convexity in its geometric form. Given a continuous linear functional $f$ on a subspace $Y \subseteq X$ with $|f(x)| \le p(x)$ for some *continuous* seminorm $p$, the Hahn–Banach theorem extends $f$ to all of $X$ while preserving the seminorm bound. If the topology has no continuous seminorms (because it is not locally convex), then there may be no nontrivial continuous linear functionals to extend.