The **strict inductive limit topology** is the canonical construction for building a locally convex topological vector space from a countable ascending chain of locally convex subspaces, each already equipped with its own topology. Rather than seeking a single family of seminorms controlling the entire space, the construction declares a set open if and only if its intersection with every piece of the chain is open in that piece. The resulting topology is the finest locally convex topology making each inclusion continuous, and it is characterised by a universal property that reduces [continuity](/page/Continuity) on the union to continuity on the pieces. The canonical application is the [test function space](/page/Test%20Function) $\mathcal{D}(\Omega)$, assembled from the Fréchet spaces $\mathcal{D}_K(\Omega)$ of smooth compactly supported [functions](/page/Function) — and the theory of [distributions](/page/Distribution) rests entirely on this inductive limit structure. [motivation] ### The Inadequacy of a Single Fréchet Topology The most tractable locally convex spaces are Fréchet spaces: complete metrizable locally convex spaces whose topology is generated by a countable family of seminorms. The [Schwartz space](/page/Schwartz%20Space) $\mathcal{S}(\mathbb{R}^n)$ and $C^\infty(\Omega)$ are both Fréchet, and their topologies are fully described by explicit, manageable seminorm families. For many spaces that arise in PDE theory and functional analysis, however, no countable seminorm family suffices, because the objects one wishes to control are not uniformly bounded by any finite collection of constraints. The difficulty becomes concrete for $\mathcal{D}(\Omega) = \bigcup_{K \subset \Omega} \mathcal{D}_K(\Omega)$, where the union ranges over compact subsets $K \subset \Omega$. Each $\mathcal{D}_K(\Omega)$ — the space of smooth functions supported in the fixed compact set $K$ — is a [Fréchet space](/page/Fr%C3%A9chet%20Space) with seminorms \begin{align*} p_{K,N}(\varphi) &= \sum_{|\alpha| \leq N} \sup_{x \in K} |\partial^\alpha \varphi(x)|. \end{align*} Passing to the union destroys metrizability: any putative Fréchet topology on $\mathcal{D}(\Omega)$ would be generated by a countable family of seminorms $\{q_j\}$, and each $q_j$ can only control test functions whose supports are confined to some compact set $K_j$. Test functions whose supports escape every $K_j$ — that is, functions that probe the behaviour of the domain arbitrarily close to $\partial \Omega$ — lie beyond the reach of the entire family $\{q_j\}$, leaving the topology unable to detect convergence for such functions. ### The Inductive Approach The resolution is to derive the topology on the union directly from the topologies already present on the pieces, rather than imposing a new one from outside. The guiding constraint is that any reasonable topology on $\mathcal{D}(\Omega)$ must make each inclusion $\mathcal{D}_K(\Omega) \hookrightarrow \mathcal{D}(\Omega)$ continuous — if the topology on the union were coarser than that of some piece, one could not detect convergence on that piece, rendering the theory of distributions incomplete. Among all locally convex topologies satisfying this constraint, the correct choice is the *finest* one: the topology that declares as many [sets](/page/Set) open as possible, subject only to local convexity and continuity of the inclusions. This is the inductive limit topology. The universal property makes this choice canonical: a linear map out of the inductive limit is continuous if and only if its restriction to each piece is continuous. Every [topological](/page/Topology) question about $\mathcal{D}(\Omega)$ thereby reduces to a family of questions about its Fréchet pieces, where the standard tools of metrizable spaces apply. This reduction is not merely a convenience — it is the mechanism by which the entire distribution calculus becomes tractable. ### The Role of Strictness The word "strict" imposes a compatibility condition between the topology of each piece and the topology it inherits from the union. Without this condition, a sequence converging in $\mathcal{D}_K(\Omega)$ might fail to converge in $\mathcal{D}(\Omega)$ with supports confined to $K$ — an internal inconsistency that would make the relationship between pieces and the ambient space difficult to control. Strictness ensures that each piece sits inside the union as a topological subspace: the topology on $\mathcal{D}_K(\Omega)$ is exactly the subspace topology inherited from $\mathcal{D}(\Omega)$, so convergence in the piece is identical to convergence in the ambient space with all supports confined to that piece. Strictness is not a formal assumption but a geometric fact in the motivating example: $\mathcal{D}_{K_n}(\Omega)$ is a closed subspace of $\mathcal{D}_{K_{n+1}}(\Omega)$ because the support condition $\mathrm{supp}(\varphi) \subseteq K_n$ is preserved under uniform [limits](/page/Limit) of all derivatives — the support of a uniform limit of functions supported in $K_n$ is again contained in $K_n$. This closedness propagates through the chain and is precisely what makes the sequential characterisation of convergence in $\mathcal{D}(\Omega)$ so clean. [/motivation] ## Definition The abstract framework applies to any countable chain of locally convex spaces. Throughout, all vector spaces are over $\mathbb{R}$; the complex case is identical. The starting point is the algebraic structure: a nested family of spaces connected by injections that are compatible with the linear and topological structure. [definition: Inductive System Of Locally Convex Spaces] An **inductive system of locally convex spaces** consists of a sequence $(V_n, \tau_n)_{n \in \mathbb{N}}$ of locally convex [topological vector spaces](/page/Topological%20Vector%20Space) together with, for each $n \in \mathbb{N}$, a continuous linear injection \begin{align*} \iota_n : V_n &\to V_{n+1}. \end{align*} For $m \leq n$, the **transition map** $\iota_{m,n} : V_m \to V_n$ is the composition \begin{align*} \iota_{m,n} &:= \iota_{n-1} \circ \cdots \circ \iota_m, \end{align*} with the convention $\iota_{n,n} := \mathrm{id}_{V_n}$. The transition maps satisfy the cocycle condition $\iota_{n,p} \circ \iota_{m,n} = \iota_{m,p}$ for all $m \leq n \leq p$. [/definition] The algebraic inductive limit makes precise the sense in which the spaces $V_n$ are glued together along the transition maps. [definition: Algebraic Inductive Limit] Let $(V_n, \iota_n)_{n \in \mathbb{N}}$ be an inductive system of locally convex spaces with transition maps $\iota_{m,n}$. The **algebraic inductive limit** is the quotient \begin{align*} V := \varinjlim_n V_n := \bigsqcup_{n=1}^\infty V_n \Big/ \sim, \end{align*} where the [equivalence relation](/page/Equivalence%20Relation) $\sim$ identifies $v \in V_m$ with $w \in V_n$ whenever there exists $N \geq m, n$ such that $\iota_{m,N}(v) = \iota_{n,N}(w)$. The **canonical injection** is the map \begin{align*} j_n : V_n &\to V \\ v &\mapsto [v]_\sim. \end{align*} [/definition] The algebraic inductive limit carries a natural vector space structure. For $[v]_\sim, [w]_\sim \in V$ with representatives $v \in V_m$ and $w \in V_n$, choose any $N \geq m, n$ and define \begin{align*} [v]_\sim + [w]_\sim &:= [\iota_{m,N}(v) + \iota_{n,N}(w)]_\sim, \\ \lambda [v]_\sim &:= [\lambda v]_\sim. \end{align*} Well-definedness of addition: suppose $v \sim v'$ and $w \sim w'$ with $v \in V_m$, $v' \in V_{m'}$, $w \in V_n$, $w' \in V_{n'}$. Choose $N$ large enough that $\iota_{m,N}(v) = \iota_{m',N}(v')$ and $\iota_{n,N}(w) = \iota_{n',N}(w')$. Then $\iota_{m,N}(v) + \iota_{n,N}(w) = \iota_{m',N}(v') + \iota_{n',N}(w')$ in $V_N$, so the equivalence class is independent of the choice of representatives and of $N$ (replacing $N$ by any larger $N'$ and applying $\iota_{N,N'}$ preserves the equality by linearity). The vector space axioms — associativity, commutativity, existence of the zero element $[0_{V_1}]_\sim$, additive inverses $-[v]_\sim = [-v]_\sim$, and distributivity — are inherited from the $V_N$. Since each $\iota_n$ is injective, the canonical injections $j_n$ are injective, and $V = \bigcup_n j_n(V_n)$. In the motivating example, $V_n = \mathcal{D}_{K_n}(\Omega)$ for an exhaustion $K_1 \subset K_2 \subset \cdots$ of $\Omega$ by compact sets with $K_j \subset \mathrm{int}(K_{j+1})$, and each $\iota_n$ is the natural inclusion (extending a function by zero outside its support). Since the inclusions are literal set-theoretic injections, $v \sim w$ if and only if $v = w$ as functions, and the quotient construction reduces to the ordinary union $\mathcal{D}(\Omega) = \bigcup_j \mathcal{D}_{K_j}(\Omega)$. The algebraic inductive limit is a vector space without a topology. Any topology on $V$ that one proposes must make each canonical injection $j_n : V_n \hookrightarrow V$ continuous, since otherwise convergence in a piece would not imply convergence in the ambient space. The inductive limit topology is the finest such topology, so that the resulting dual space is as large as possible — ensuring that the maximum number of linear functionals are continuous. [definition: Inductive Limit Topology] Let $(V_n, \iota_n)_{n \in \mathbb{N}}$ be an inductive system of locally convex spaces with algebraic inductive limit $V$ and canonical injections $j_n : V_n \to V$. The **inductive limit topology** on $V$ is the finest locally convex topology on $V$ such that $j_n$ is continuous for every $n \in \mathbb{N}$. [/definition] The inductive limit topology admits a concrete description in terms of neighbourhoods: a convex balanced set $U \subseteq V$ is a neighbourhood of the origin in the inductive limit topology if and only if $j_n^{-1}(U)$ is a neighbourhood of the origin in $(V_n, \tau_n)$ for every $n \in \mathbb{N}$. This makes it explicit that the topology is generated by the family of all seminorms $p$ on $V$ whose restriction $p \circ j_n$ is continuous on $(V_n, \tau_n)$ for every $n$. The inductive limit topology is uniquely determined by the maximality condition together with local convexity. However, there is a potential pathology: the subspace topology that the inductive limit induces on $V_n$ might be strictly coarser than $\tau_n$, meaning some [open sets](/page/Open%20Set) of $V_n$ fail to appear as intersections with open sets of $V$. The strict condition ensures this cannot happen. [definition: Strict Inductive System] An inductive system $(V_n, \iota_n)_{n \in \mathbb{N}}$ is **strict** if for every $n \in \mathbb{N}$: 1. The image $\iota_n(V_n)$ is a closed subspace of $(V_{n+1}, \tau_{n+1})$. 2. The topology $\tau_n$ coincides with the subspace topology that $\iota_n(V_n)$ inherits from $(V_{n+1}, \tau_{n+1})$. [/definition] [definition: Strict Inductive Limit] The **strict inductive limit** of a strict inductive system $(V_n, \iota_n)_{n \in \mathbb{N}}$ is the algebraic inductive limit $V = \varinjlim_n V_n$ equipped with the inductive limit topology. [/definition] The closedness of $V_n$ in $V_{n+1}$ is the operative condition: it ensures that the embedding $j_n : V_n \to V$ is not merely a continuous injection but a topological embedding, so that the topology $\tau_n$ coincides with the subspace topology of $V_n$ in $V$. This is the content of the embedding theorem developed below. [example: The Standard Strict Inductive System For D] Fix an open set $\Omega \subseteq \mathbb{R}^n$ and choose an exhaustion by compact sets $K_1 \subset K_2 \subset \cdots$ with $K_j \subset \mathrm{int}(K_{j+1})$ and $\bigcup_j K_j = \Omega$. For each $j \in \mathbb{N}$, the space $\mathcal{D}_{K_j}(\Omega) = \{\varphi \in C^\infty(\Omega) : \mathrm{supp}(\varphi) \subseteq K_j\}$ is a Fréchet space with the seminorms \begin{align*} p_{K_j, N}(\varphi) &= \sum_{|\alpha| \leq N} \sup_{x \in K_j} |\partial^\alpha \varphi(x)|, \quad N \in \mathbb{N}_0. \end{align*} Each inclusion $\iota_j : \mathcal{D}_{K_j}(\Omega) \hookrightarrow \mathcal{D}_{K_{j+1}}(\Omega)$ is continuous because $p_{K_{j+1}, N}(\varphi) \geq p_{K_j, N}(\varphi)$ for all $\varphi \in \mathcal{D}_{K_j}(\Omega)$ (the supremum is taken over a larger set but the function vanishes outside $K_j$, so the values coincide). The image $\iota_j(\mathcal{D}_{K_j}(\Omega))$ is closed in $\mathcal{D}_{K_{j+1}}(\Omega)$: if $\varphi_m \in \mathcal{D}_{K_j}(\Omega)$ and $\varphi_m \to \varphi$ in $\mathcal{D}_{K_{j+1}}(\Omega)$, then $\partial^\alpha \varphi_m \to \partial^\alpha \varphi$ uniformly on $K_{j+1}$ for every $\alpha$. For any $x \in K_{j+1} \setminus K_j$, each $\varphi_m(x) = 0$, so $\varphi(x) = 0$ by pointwise convergence. Hence $\mathrm{supp}(\varphi) \subseteq K_j$ and $\varphi \in \mathcal{D}_{K_j}(\Omega)$. Finally, the subspace topology on $\mathcal{D}_{K_j}(\Omega)$ from $\mathcal{D}_{K_{j+1}}(\Omega)$ coincides with $\tau_j$: both topologies are generated by seminorms of the form $\sum_{|\alpha| \leq N} \sup |\partial^\alpha \varphi|$, and for functions supported in $K_j$ the supremum over $K_{j+1}$ equals the supremum over $K_j$. The strict inductive limit $\mathcal{D}(\Omega) = \bigcup_j \mathcal{D}_{K_j}(\Omega)$ is independent of the choice of exhaustion — different exhaustions yield the same topology on $\mathcal{D}(\Omega)$. This independence follows from the universal property: if two exhaustions produce inductive limit topologies $\tau$ and $\tau'$, the identity map is continuous in both directions (since each $\mathcal{D}_K$ for any compact $K$ is contained in some piece of either exhaustion), forcing $\tau = \tau'$. [/example] ## The Universal Property The inductive limit construction is not merely one possible topology on $V$; it is the *right* topology, characterised by a universal property. The universal property asserts that continuity on the entire space $V$ is equivalent to continuity on each piece simultaneously, and this equivalence is what makes the inductive limit a categorical colimit in the category of locally convex spaces. [quotetheorem:706] The proof is a direct unfolding of definitions. If $T$ is continuous and $U \subseteq W$ is open, then for each $n$ the preimage $(T \circ j_n)^{-1}(U) = T^{-1}(U) \cap V_n$ is open in $V_n$ because $T^{-1}(U)$ is open in $V$ and $j_n$ is continuous. Conversely, suppose each $T \circ j_n$ is continuous. For any open $U \subseteq W$ and any $n \in \mathbb{N}$, the set $T^{-1}(U) \cap V_n = (T \circ j_n)^{-1}(U)$ is open in $(V_n, \tau_n)$. By the characterisation of open sets in the inductive limit topology, $T^{-1}(U)$ is open in $V$, so $T$ is continuous. This theorem is the engine of [distribution](/page/Distribution) theory. To verify that a linear functional $T : \mathcal{D}(\Omega) \to \mathbb{R}$ is continuous — and hence defines a distribution — one checks continuity of $T|_{\mathcal{D}_{K_n}(\Omega)}$ for each compact $K_n$ in a fixed exhaustion. Each $\mathcal{D}_{K_n}(\Omega)$ is Fréchet, so continuity there has the concrete, checkable criterion: there exist constants $C_K > 0$ and $N_K \in \mathbb{N}_0$ such that \begin{align*} |T(\varphi)| &\leq C_K \sum_{|\alpha| \leq N_K} \sup_{x \in K} |\partial^\alpha \varphi(x)| \quad \text{for all } \varphi \in \mathcal{D}_K(\Omega). \end{align*} The [Characterisation of Distributions](/theorems/449) is exactly this reduction. The universal property also governs the topology on spaces of distributions: a linear map $T : \mathcal{D}'(\Omega) \to W$ is continuous in the [weak* topology](/page/Weak*%20Topology) if and only if $\varphi \mapsto T(u)(\varphi)$ is continuous on each $\mathcal{D}_K(\Omega)$ for every $u$, reducing questions about the distribution space to questions about its Fréchet pieces. [example: Continuity Of Differentiation On D] The partial derivative operator \begin{align*} \partial^\alpha : \mathcal{D}(\Omega) &\to \mathcal{D}(\Omega) \\ \varphi &\mapsto \partial^\alpha \varphi \end{align*} is continuous for every multi-index $\alpha$. By the universal property, it suffices to verify continuity of $\partial^\alpha|_{\mathcal{D}_{K_j}(\Omega)}$ for each $j$. Since $\partial^\alpha \varphi$ is supported in $K_j$ whenever $\varphi$ is ([differentiation](/page/Derivative) does not enlarge support), the restricted map lands in $\mathcal{D}_{K_j}(\Omega)$, and the estimate \begin{align*} p_{K_j, N}(\partial^\alpha \varphi) &= \sum_{|\beta| \leq N} \sup_{x \in K_j} |\partial^{\beta + \alpha} \varphi(x)| \leq p_{K_j, N + |\alpha|}(\varphi) \end{align*} shows that $\partial^\alpha|_{\mathcal{D}_{K_j}(\Omega)}$ is continuous as a map between Fréchet spaces. The universal property then gives global continuity. [/example] ## Topological Properties ### The Embedding Property A basic concern with inductive limits is whether the pieces sit inside the union in a well-behaved way. In a non-strict inductive system, the inductive limit topology can be strictly coarser than $\tau_n$ on $V_n$: some open sets of $V_n$ might not arise as intersections with open sets of $V$, meaning the inclusion $j_n$ is continuous but fails to be a topological embedding. Such a failure would decouple the topology of a piece from the ambient topology, making the relationship between local and global convergence unpredictable. The embedding property asserts that strictness prevents this failure entirely. [quotetheorem:707] The core of the argument is a layer-by-layer extension of neighbourhoods. Fix $n$ and let $U_n$ be a convex balanced neighbourhood of the origin in $(V_n, \tau_n)$. The strictness condition guarantees that $U_n = V_n \cap U_{n+1}$ for some convex balanced neighbourhood $U_{n+1}$ of the origin in $V_{n+1}$ (since $\tau_n$ is the subspace topology from $V_{n+1}$). Repeating this for $U_{n+1}$ in $V_{n+2}$, and continuing inductively, one constructs a compatible system of neighbourhoods $U_n \subseteq U_{n+1} \subseteq U_{n+2} \subseteq \cdots$ with $U_k \cap V_k = U_k$ for each $k \geq n$. The set $U = \bigcup_{k \geq n} U_k$ is then a convex balanced subset of $V$ satisfying $U \cap V_k \in \tau_k$ for every $k$, and by definition $U$ is open in the inductive limit topology with $U \cap V_n = U_n$. This shows that every $\tau_n$-neighbourhood arises as the intersection of an inductive-limit-open set with $V_n$, so the subspace topology on $V_n$ is at least as fine as $\tau_n$. The reverse inclusion — the subspace topology is no finer than $\tau_n$ — holds because $j_n$ is continuous by definition. For $\mathcal{D}(\Omega)$, the embedding property means that a sequence of test functions supported in a fixed compact $K$ converges to zero in $\mathcal{D}_K(\Omega)$ if and only if it converges to zero in $\mathcal{D}(\Omega)$. The two notions of convergence on $\mathcal{D}_K(\Omega)$ agree completely. ### Hausdorff Separation and Completeness The Hausdorff property of a strict inductive limit follows readily from the embedding property and the Hausdorff property of the pieces. Two distinct points $x, y \in V$ both belong to some $V_n$ (since the union is countable and increasing). The space $(V_n, \tau_n)$ is Hausdorff, so there exist disjoint $\tau_n$-open sets separating $x$ and $y$. The embedding property promotes these to open sets in $V$, giving Hausdorff separation in the inductive limit. Completeness is a deeper property that requires the full strength of the closedness condition and uses a fundamental structural lemma about bounded sets. [quotetheorem:709] This "bounded sets are eventually bounded" property is the most important structural fact about strict inductive limits beyond the embedding theorem. It says that every bounded set is captured by a single piece of the chain — no bounded set can straddle infinitely many pieces. The proof proceeds by contradiction. Suppose $B$ is bounded in $V$ but not contained in any $V_n$. Then for each $n$ there exists $x_n \in B \setminus V_n$. Since $V_n$ is closed in $V_{n+1}$ and hence closed in $V$ (by the embedding property), the Hahn-Banach separation theorem in $V_{n+1}$ produces a continuous linear functional $f_n$ on $V$ vanishing on $V_n$ with $f_n(x_n)$ arbitrarily large. Specifically, one constructs a continuous seminorm $p$ on $V$ with $p(x_n) \to \infty$, contradicting the boundedness of $B$ (which requires $p(B)$ to be bounded for every continuous seminorm $p$). The boundedness lemma has an immediate application to completeness: it is the key ingredient in showing that Cauchy nets cannot escape to infinity along the chain. [quotetheorem:711] Completeness here means that every Cauchy filter on $V$ converges — the appropriate notion for non-metrizable spaces, since [Cauchy sequences](/page/Cauchy%20Sequence) do not suffice to characterise completeness beyond the metrizable setting. The argument uses the bounded-sets lemma as follows. A Cauchy filter $\mathcal{F}$ on $V$ contains a bounded set $B$ (every Cauchy filter is eventually contained in a translate of any neighbourhood, so by taking a decreasing sequence of neighbourhoods one obtains a bounded member). By the bounded-sets lemma, $B \subseteq V_n$ for some $n$, and the trace $\mathcal{F} \cap V_n = \{F \cap V_n : F \in \mathcal{F}\}$ is a Cauchy filter on $V_n$. Since $V_n$ is Fréchet and hence complete, $\mathcal{F} \cap V_n$ converges to some $x \in V_n$. The embedding property ensures that convergence in $V_n$ implies convergence in $V$, so $\mathcal{F} \to x$ in $V$. Completeness may seem surprising given that the space is non-metrizable, since one typically encounters completeness in the metric setting. For $\mathcal{D}(\Omega)$, completeness means that every Cauchy net of test functions converges in $\mathcal{D}(\Omega)$ — a fundamental stability property underpinning the theory of distributions. ### Non-Metrizability Despite Hausdorff separation and completeness, strict inductive limits of Fréchet spaces with strictly increasing pieces are never metrizable. This is not a deficiency but a structural necessity: the non-metrizability reflects the fact that no single countable family of seminorms can simultaneously control functions with arbitrarily large support. [quotetheorem:713] The proof is a Baire category argument. If $V$ were metrizable, it would be a complete metrizable space (by the completeness theorem above) and hence a Baire space. The decomposition $V = \bigcup_n V_n$ expresses $V$ as a countable union. Each $V_n$ is closed in $V$ (the closedness of $V_n$ in $V_{n+1}$, together with a straightforward induction, gives closedness in $V$). By the [Baire category theorem](/theorems/630), some $V_{n_0}$ has non-empty interior in $V$: there exists a non-empty open set $U \subseteq V$ with $U \subseteq V_{n_0}$. But in a topological vector space, any non-empty open set generates the entire space under scalar multiplication and translation: $V = \bigcup_{k=1}^\infty k \cdot U \subseteq V_{n_0}$, since $V_{n_0}$ is a subspace and hence closed under scalar multiplication. This contradicts $V_{n_0} \subsetneq V_{n_0 + 1} \subseteq V$. [example: Non-Metrizability Of D On R] Let $\Omega = \mathbb{R}$ and $K_n = [-n, n]$. For each $n \in \mathbb{N}$, construct a smooth bump function $\psi_n \in C_c^\infty(\mathbb{R})$ with $\mathrm{supp}(\psi_n) \subseteq (n, n+1)$ and $\|\psi_n\|_\infty = 1$ (e.g., by [mollifying](/page/Standard%20Mollifier) $\mathbb{1}_{[n+1/4, n+3/4]}$). Then $\psi_n \in \mathcal{D}_{K_{n+1}}(\mathbb{R}) \setminus \mathcal{D}_{K_n}(\mathbb{R})$. Consider the sequence $(\varepsilon_n \psi_n)_{n \in \mathbb{N}}$ for any $\varepsilon_n \to 0$. For any fixed seminorm $p_{K_m, N}$ (controlling functions supported in $K_m$), one has $p_{K_m, N}(\varepsilon_n \psi_n) = 0$ for all $n > m$, since $\mathrm{supp}(\psi_n) \cap K_m = \emptyset$. So $\varepsilon_n \psi_n \to 0$ in every individual seminorm of the form $p_{K_m, N}$. However, the supports of the $\psi_n$ escape every fixed compact set, so $(\varepsilon_n \psi_n)$ fails the support condition of the [Sequential Characterisation of Convergence in the Test Function Space](/theorems/448): there is no single compact $K$ containing all the supports. The sequence therefore does not converge to zero in $\mathcal{D}(\mathbb{R})$. If the topology on $\mathcal{D}(\mathbb{R})$ were metrizable, it would be determined by a countable family of seminorms, and the above computation shows $\varepsilon_n \psi_n \to 0$ in each of them — yet the sequence does not converge. No countable collection of seminorms can detect the divergence, confirming non-metrizability directly. [/example] [remark: Bornological Character Of LF-Spaces] The non-metrizability of strict inductive limits means that many classical tools of Fréchet space theory — the [open mapping theorem](/theorems/631), the [closed graph theorem](/theorems/217), the Banach-Steinhaus theorem in their standard Banach/Fréchet formulations — do not apply directly. However, LF-spaces satisfy a substitute: they are *bornological* (every convex balanced set that absorbs all bounded sets is a neighbourhood) and *barrelled* (every barrel — closed, convex, balanced, absorbing set — is a neighbourhood). The barrelled property ensures that the Banach-Steinhaus theorem holds for LF-spaces: a pointwise bounded family of continuous [linear maps](/page/Linear%20Map) on an LF-space is equicontinuous. [/remark] ## LF-Spaces and Sequential Continuity Strict inductive limits of Fréchet spaces arise often enough to merit their own name and their own theory. The key structural feature of this class — the equivalence of sequential continuity and full topological continuity — is the property that makes distribution theory accessible in practice, despite the non-metrizable topology. In a general locally convex space, sequential continuity does not imply topological continuity: a linear map can preserve limits of [sequences](/page/Sequence) without being continuous in the full sense (preserving limits of nets and filters). In Fréchet spaces, the two properties coincide because the topology is metrizable and hence determined by sequences. The remarkable fact is that LF-spaces inherit this sequential character despite their non-metrizability. [definition: LF-Space] An **LF-space** (limit of Fréchet spaces) is a locally convex topological vector space that is isomorphic, as a locally convex space, to the strict inductive limit of a countable sequence of Fréchet spaces $(V_n, \tau_n)_{n \in \mathbb{N}}$ with $V_n \subsetneq V_{n+1}$. [/definition] The test function space $\mathcal{D}(\Omega)$ is the canonical LF-space. The [Schwartz space](/page/Schwartz%20Space) $\mathcal{S}(\mathbb{R}^n)$, by contrast, is Fréchet — it is *not* an LF-space, because it carries a countable seminorm family. The distinction is essential: $\mathcal{D}(\Omega)$ is non-metrizable, while $\mathcal{S}(\mathbb{R}^n)$ is metrizable. [quotetheorem:715] The proof combines the bounded-sets lemma with the universal property. Suppose $T$ is sequentially continuous. To show full continuity, by the universal property it suffices to show that $T|_{V_n}$ is continuous for each $n$. Each $V_n$ is Fréchet and hence metrizable, so sequential continuity of $T|_{V_n}$ is equivalent to continuity of $T|_{V_n}$. It therefore suffices to show that $T|_{V_n}$ is sequentially continuous. Let $(x_k)_{k=1}^\infty$ be a sequence in $V_n$ with $x_k \to 0$ in $V_n$. By the embedding property, $x_k \to 0$ in $V$. By the sequential continuity of $T$ on $V$, $T(x_k) \to 0$ in $W$. Hence $T|_{V_n}$ is sequentially continuous on the Fréchet space $V_n$, and therefore continuous. This theorem — combined with the [Sequential Characterisation of Convergence in the Test Function Space](/theorems/448) — reduces the entire abstract topology of $\mathcal{D}(\Omega)$ to two concrete conditions on sequences. To verify that a linear functional $T : \mathcal{D}(\Omega) \to \mathbb{R}$ is a distribution, one need only check: whenever $\varphi_k \to \varphi$ in $\mathcal{D}(\Omega)$ (meaning the supports are uniformly bounded and all derivatives converge uniformly), one has $T(\varphi_k) \to T(\varphi)$. The [Characterisation of Distributions](/theorems/449) gives the explicit seminorm estimate that this sequential condition entails. One never needs to reason about nets or Cauchy filters; sequences with uniformly bounded supports and [uniformly convergent](/page/Uniform%20Convergence) derivatives suffice to determine every topological datum of $\mathcal{D}(\Omega)$. [example: The Dirac Delta Is A Distribution Via Sequential Continuity] Fix $x_0 \in \Omega$ and define the Dirac delta at $x_0$ by \begin{align*} \delta_{x_0} : \mathcal{D}(\Omega) &\to \mathbb{R} \\ \varphi &\mapsto \varphi(x_0). \end{align*} Linearity is immediate. To verify that $\delta_{x_0}$ is a distribution, the sequential continuity criterion suffices: if $\varphi_k \to \varphi$ in $\mathcal{D}(\Omega)$, then by the sequential characterisation there exists a compact $K$ with all supports in $K$ and $\varphi_k \to \varphi$ uniformly on $K$. In particular $\varphi_k(x_0) \to \varphi(x_0)$, so $\delta_{x_0}(\varphi_k) \to \delta_{x_0}(\varphi)$. The sequential continuity theorem then gives full continuity of $\delta_{x_0}$ with respect to the inductive limit topology. Alternatively, the estimate from the [Characterisation of Distributions](/theorems/449) is simply $|\delta_{x_0}(\varphi)| = |\varphi(x_0)| \leq \sup_{x \in K} |\varphi(x)| = p_{K, 0}(\varphi)$ for any compact $K$ containing $x_0$ — an order-zero estimate with $C_K = 1$ and $N_K = 0$. [/example] ## The Canonical Example: Test Functions The [test function space](/page/Test%20Function) $\mathcal{D}(\Omega)$ is the LF-space motivating the entire construction. For any exhaustion $K_1 \subset K_2 \subset \cdots$ of $\Omega$ by compact sets with $K_j \subset \mathrm{int}(K_{j+1})$ and $\bigcup_j K_j = \Omega$, the spaces $\mathcal{D}_{K_j}(\Omega)$ form a strict inductive system of Fréchet spaces, as verified in the example above. The strict inductive limit topology on $\mathcal{D}(\Omega)$ is the unique locally convex topology that makes every inclusion $\mathcal{D}_{K_j}(\Omega) \hookrightarrow \mathcal{D}(\Omega)$ a topological embedding and is maximal with respect to this property. The general theorems of this page specialise to give the complete topological picture of $\mathcal{D}(\Omega)$. The space is Hausdorff, complete, non-metrizable, bornological, and barrelled. The sequential characterisation of convergence — which describes convergence in $\mathcal{D}(\Omega)$ by two concrete conditions on supports and derivatives — follows from the embedding property and the definition of the inductive limit topology. [quotetheorem:448] The power of this characterisation is that it replaces the abstract inductive limit topology with two checkable conditions: uniform support containment and uniform convergence of all derivatives. No reference to the inductive limit machinery is needed to *use* the topology in practice — only to *prove* that the sequential description is correct. The forward direction (convergence in the inductive limit implies the two conditions) uses the embedding property: if $\varphi_k \to \varphi$ in $\mathcal{D}(\Omega)$, then the bounded-sets lemma forces $\{\varphi_k\}$ into some $\mathcal{D}_{K_j}(\Omega)$, and the embedding property gives convergence there, which is precisely the two conditions. The reverse direction uses the universal property: convergence in $\mathcal{D}_{K_j}(\Omega)$ implies convergence tested against any continuous linear functional, hence convergence in $\mathcal{D}(\Omega)$. [quotetheorem:449] The equivalence $(1) \Leftrightarrow (2)$ is the sequential continuity theorem applied to the special case $W = \mathbb{R}$. The equivalence $(1) \Leftrightarrow (3)$ unwinds continuity on each Fréchet piece into a concrete seminorm estimate: continuity of $T|_{\mathcal{D}_K(\Omega)}$ means precisely that $T$ is bounded by some finite sum of supremum seminorms on $\mathcal{D}_K(\Omega)$. The order $N_K$ may vary with $K$ — a distribution need not have a uniform bound on the order of derivatives it consumes. Distributions for which a uniform bound exists are called distributions of *finite order*. ### Independence Of Exhaustion A natural question is whether the topology on $\mathcal{D}(\Omega)$ depends on the choice of exhaustion $K_1 \subset K_2 \subset \cdots$. The answer is no: any two exhaustions produce the same inductive limit topology, so $\mathcal{D}(\Omega)$ is well-defined as a topological vector space. The proof uses the universal property. Let $\{K_j\}$ and $\{L_j\}$ be two exhaustions and let $\tau$ and $\sigma$ be the corresponding inductive limit topologies. Every $\mathcal{D}_{K_j}(\Omega)$ is contained in some $\mathcal{D}_{L_m}(\Omega)$ (choose $m$ large enough that $K_j \subseteq L_m$), and the inclusion $\mathcal{D}_{K_j}(\Omega) \hookrightarrow (\mathcal{D}(\Omega), \sigma)$ factors through $\mathcal{D}_{L_m}(\Omega) \hookrightarrow (\mathcal{D}(\Omega), \sigma)$, which is continuous. The universal property of $\tau$ gives that the identity map $(\mathcal{D}(\Omega), \tau) \to (\mathcal{D}(\Omega), \sigma)$ is continuous. Reversing the roles yields continuity in both directions, so $\tau = \sigma$. ## References - Rudin, *Functional Analysis* (1991). - Trèves, *Topological Vector Spaces, Distributions and Kernels* (1967). - Schaefer and Wolff, *Topological Vector Spaces* (1999). - Hörmander, *The Analysis of Linear Partial Differential Operators I* (1983).