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. The space $L^p(\Omega)$ for $0 < p < 1$ is a concrete example: it is a complete metrisable TVS (an $F$-space), but it is not locally convex, and its dual is trivial — the only continuous linear functional is $f = 0$. This is why $L^p$ spaces with $p < 1$ do not appear in PDE theory: without a dual space, there is no theory of weak solutions or distributions. ## Examples The power of the TVS framework lies in its ability to accommodate spaces of very different character under a single theory. The following examples are listed in order of increasing generality. [example:Normed Spaces And Banach Spaces] Every normed space $(X, \|\cdot\|)$ is a locally convex TVS with the topology generated by the single seminorm $p = \|\cdot\|$. Since a single norm is separating, the topology is Hausdorff. The basic neighbourhoods at the origin are the open balls $B(0, \varepsilon) = \{x : \|x\| < \varepsilon\}$. A normed space is a Banach space if it is complete (every [Cauchy sequence](/page/Cauchy%20Sequence) converges). The spaces $L^p(\Omega)$ for $1 \le p \le \infty$, the [Sobolev spaces](/page/Sobolev%20Spaces) $W^{k,p}(\Omega)$, and the [Hilbert spaces](/page/Hilbert%20Space) are all [Banach spaces](/page/Banach%20Space). [/example] [example:Frechet Spaces] A **Fréchet space** is a complete, metrisable, locally convex TVS. Equivalently, it is a complete locally convex space whose topology is generated by a *countable* separating family of seminorms $\{p_n\}_{n=1}^\infty$. The metrisability follows from countability: the translation-invariant metric \begin{align*} d(x, y) := \sum_{n=1}^\infty \frac{1}{2^n} \frac{p_n(x - y)}{1 + p_n(x - y)} \end{align*} generates the same topology as the seminorms. The prototype is $C^\infty(\Omega)$ for an open $\Omega \subseteq \mathbb{R}^n$: choose an exhaustion $K_1 \subset K_2 \subset \dots$ of $\Omega$ by compact sets and define \begin{align*} p_{k,m}(f) := \sup_{\substack{x \in K_k \\ |\alpha| \le m}} |D^\alpha f(x)|. \end{align*} The family $\{p_{k,m}\}_{k, m \in \mathbb{N}}$ is countable and separating, and $C^\infty(\Omega)$ is complete under the resulting metric. A sequence $f_j \to f$ in this topology if and only if $D^\alpha f_j \to D^\alpha f$ uniformly on every compact subset of $\Omega$, for every multi-index $\alpha$. The Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is another Fréchet space, with seminorms \begin{align*} \|f\|_{\alpha, \beta} := \sup_{x \in \mathbb{R}^n} |x^\alpha D^\beta f(x)| \end{align*} indexed by multi-indices $\alpha, \beta$. A function $f$ is in $\mathcal{S}(\mathbb{R}^n)$ if and only if all these seminorms are finite — meaning $f$ and all its derivatives decay faster than any polynomial. [/example] The next example is the most important non-metrisable locally convex space in analysis, and the reason the full generality of the TVS framework is needed. [example:The Space Of Test Functions] The [test function space](/page/Test%20Function) $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ requires special care. For each compact $K \subset \Omega$, let $\mathcal{D}_K(\Omega) := \{f \in C_c^\infty(\Omega) : \operatorname{supp} f \subseteq K\}$. Each $\mathcal{D}_K(\Omega)$ is a [Fréchet space](/page/Fr%C3%A9chet%20Space) (with the seminorms $\sup_{|\alpha| \le m} \sup_{x \in K} |D^\alpha f(x)|$). The space $\mathcal{D}(\Omega) = \bigcup_K \mathcal{D}_K(\Omega)$ is topologised as the **strict inductive limit** of the $\mathcal{D}_K(\Omega)$: a set $U \subseteq \mathcal{D}(\Omega)$ is open if and only if $U \cap \mathcal{D}_K(\Omega)$ is open in $\mathcal{D}_K(\Omega)$ for every compact $K$. This topology is locally convex (one can show the inductive limit of locally convex spaces is locally convex) but it is *not metrisable* — there is no countable family of seminorms that generates it. The failure of metrisability means that sequential characterisations of convergence and continuity can be misleading: a linear map $T: \mathcal{D}(\Omega) \to Y$ is continuous if and only if $T|_{\mathcal{D}_K(\Omega)}$ is continuous for every compact $K$, which is a stronger condition than sequential continuity (though for linear maps on $\mathcal{D}(\Omega)$, the two conditions happen to be equivalent). [/example] [example:Weak And Weak Star Topologies] Let $X$ be a Banach space. The **weak topology** on $X$ is the locally convex topology generated by the seminorms $p_f(x) := |f(x)|$ indexed by $f \in X^*$. A net $x_i \rightharpoonup x$ if and only if $f(x_i) \to f(x)$ for every $f \in X^*$. The [weak* topology](/page/Weak*%20Topology) on $X^*$ is the locally convex topology generated by the seminorms $p_x(f) := |f(x)|$ indexed by $x \in X$. A net $f_i \overset{*}{\rightharpoonup} f$ if and only if $f_i(x) \to f(x)$ for every $x \in X$. Both are Hausdorff locally convex topologies that are strictly weaker than the norm topology (when $X$ is infinite-dimensional). They are metrisable on bounded subsets when $X$ (respectively $X^*$) is separable, but not globally metrisable in infinite dimensions. [/example] ## Continuity of Linear Maps In a normed space, a linear map is continuous if and only if it is bounded: $\|Tx\| \le C\|x\|$. In a locally convex space, the characterisation replaces the single norm bound with a seminorm bound. [definition:Continuous Linear Map Between Locally Convex Spaces] Let $(X, \mathcal{P})$ and $(Y, \mathcal{Q})$ be locally convex TVS with generating seminorm families $\mathcal{P} = \{p_\alpha\}_{\alpha \in A}$ and $\mathcal{Q} = \{q_\beta\}_{\beta \in B}$. A linear map $T: X \to Y$ is **continuous** if and only if for every $\beta \in B$, there exist finitely many indices $\alpha_1, \dots, \alpha_n \in A$ and a constant $C > 0$ such that \begin{align*} q_\beta(Tx) \le C \max\{p_{\alpha_1}(x), \dots, p_{\alpha_n}(x)\} \quad \text{for all } x \in X. \end{align*} Equivalently, $T$ is continuous if and only if for every continuous seminorm $q$ on $Y$, the composition $q \circ T$ is a continuous seminorm on $X$. [/definition] This reduces to the familiar boundedness condition in the normed case (where both $\mathcal{P}$ and $\mathcal{Q}$ are singletons). For Fréchet spaces, the continuity condition becomes: for every seminorm $q_m$ on $Y$, there exist $N \in \mathbb{N}$ and $C > 0$ such that $q_m(Tx) \le C \max_{n \le N} p_n(x)$. The **dual space** of a locally convex TVS $X$ is the vector space $X'$ of all continuous linear functionals $f: X \to \mathbb{R}$. A linear functional $f: X \to \mathbb{R}$ is continuous if and only if there exist finitely many seminorms $p_{\alpha_1}, \dots, p_{\alpha_n} \in \mathcal{P}$ and $C > 0$ with $|f(x)| \le C \max_k p_{\alpha_k}(x)$ for all $x$. In the normed case, this recovers $|f(x)| \le C\|x\|$, i.e. $f \in X^*$. ### The Initial Topology Construction Many of the topologies encountered in analysis arise from a single abstract construction: given a vector space $X$, a collection of TVS $\{Y_\alpha\}_{\alpha \in A}$, and linear maps $T_\alpha: X \to Y_\alpha$, the **initial topology** (or **projective topology**) on $X$ is the coarsest topology making all the $T_\alpha$ continuous. If each $Y_\alpha$ is locally convex with seminorms $\{q_{\alpha, \beta}\}_\beta$, the initial topology on $X$ is generated by the seminorms $\{q_{\alpha, \beta} \circ T_\alpha\}_{\alpha, \beta}$, and it is locally convex. This construction unifies several fundamental examples. The weak topology on a Banach space $X$ is the initial topology on $X$ with respect to the evaluation maps $\{\text{ev}_f : X \to \mathbb{R}\}_{f \in X^*}$, where $\text{ev}_f(x) = f(x)$. The weak* topology on $X^*$ is the initial topology on $X^*$ with respect to $\{\text{ev}_x : X^* \to \mathbb{R}\}_{x \in X}$. The topology on $C^\infty(\Omega)$ is the initial topology with respect to the restriction maps $\{R_K: C^\infty(\Omega) \to C(K)\}_{K \text{ compact}}$ composed with the derivative operators. Recognising a topology as an initial topology immediately tells you what the continuous linear functionals are (composites of the defining maps with continuous functionals on the target spaces) and what net convergence means (convergence in every component). ## Completeness and Metrisability Two properties determine the analytical power of a locally convex space: whether it is *complete* (so that limits of Cauchy nets exist) and whether it is *metrisable* (so that [sequences](/page/Sequence) suffice for topological arguments). [definition:Completeness In A Locally Convex Space] A net $\{x_i\}_{i \in I}$ in a locally convex TVS $(X, \mathcal{P})$ is **Cauchy** if for every seminorm $p_\alpha \in \mathcal{P}$ and every $\varepsilon > 0$, there exists $i_0 \in I$ such that $p_\alpha(x_i - x_j) < \varepsilon$ for all $i, j \ge i_0$. The space $X$ is **complete** if every Cauchy net converges to an element of $X$. A complete locally convex TVS is also called a **complete locally convex space**. [/definition] Metrisability is controlled by countability: a locally convex TVS is metrisable if and only if its topology can be generated by a *countable* family of seminorms. In this case, the translation-invariant metric $d(x,y) = \sum_n 2^{-n} p_n(x-y)/(1 + p_n(x-y))$ generates the topology, and completeness can be checked using sequences rather than nets. A **Fréchet space** is a complete metrisable locally convex TVS — equivalently, a complete locally convex space whose topology is generated by a countable separating family of seminorms. The analytical significance of Fréchet spaces is that the [Baire category theorem](/theorems/630) applies (they are complete [metric spaces](/page/Metric%20Space)), so the [open mapping theorem](/theorems/631), the [closed graph theorem](/theorems/217), and the [uniform boundedness principle](/theorems/549) all hold for continuous linear maps between Fréchet spaces. [remark:Beyond Frechet Spaces] The space $\mathcal{D}(\Omega)$ is complete (as an inductive limit of Fréchet spaces) but not metrisable. It is an example of an **LF-space** — a countable strict inductive limit of Fréchet spaces. For LF-spaces, the open mapping theorem and closed graph theorem still hold (this is a theorem of Grothendieck and De Wilde), but the proofs are more delicate than in the Fréchet case because the Baire category theorem does not apply directly. [/remark] ## The Hierarchy of Topological Vector Spaces The spaces encountered in analysis sit in a hierarchy of increasing generality. Each level sacrifices some analytical tools but gains the flexibility to accommodate a wider class of spaces. At the most restricted level sit the **Banach spaces**: complete normed spaces. Here the full power of classical functional analysis is available — the Hahn–Banach theorem, the open mapping theorem, the closed graph theorem, the [uniform boundedness principle](/theorems/549), and the spectral theory of [bounded](/page/Linear%20Operators%20on%20Banach%20Spaces) and [self-adjoint](/page/Self-Adjoint%20Operators) operators. One step up are the **Fréchet spaces**: complete, metrisable, locally convex. The norm is replaced by a countable family of seminorms, but the Baire category theorem still holds, so the main pillars of functional analysis survive. The spaces $C^\infty(\Omega)$ and $\mathcal{S}(\mathbb{R}^n)$ live here. Next are the **LF-spaces** (countable strict inductive limits of Fréchet spaces), of which $\mathcal{D}(\Omega)$ is the prototype. Metrisability is lost, but completeness and the closed graph theorem survive. At the broadest useful level are the **complete locally convex spaces**. The Hahn–Banach theorem and the theory of duality (weak and weak* topologies, polar sets, the Mackey–Arens theorem) work in full generality here. The [weak* topology](/page/Weak*%20Topology) on the dual of any locally convex space is of this type. Beyond local convexity, the theory degenerates rapidly: the dual may be trivial, and the Hahn–Banach theorem fails. [example:Where Common Spaces Sit In The Hierarchy] The following table locates the main function spaces within the hierarchy: | Space | Type | Seminorms | Metrisable? | Complete? | |---|---|---|---|---| | $\mathbb{R}^n$ | Banach (Hilbert) | Euclidean norm | Yes | Yes | | $L^p(\Omega)$, $1 \le p \le \infty$ | Banach | $L^p$ norm | Yes | Yes | | $W^{k,p}(\Omega)$ | Banach | Sobolev norm | Yes | Yes | | $C^\infty(\Omega)$ | Fréchet | $\sup_{K, |\alpha| \le m} \|D^\alpha f\|_{C(K)}$ | Yes | Yes | | $\mathcal{S}(\mathbb{R}^n)$ | Fréchet | $\sup |x^\alpha D^\beta f|$ | Yes | Yes | | $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ | LF-space | Inductive limit | No | Yes | | $(X^*, \text{weak*})$, $X$ Banach | Locally convex | $|f(x)|$, $x \in X$ | On bounded subsets if $X$ separable | Yes (if $X$ complete) | | $L^p(\Omega)$, $0 < p < 1$ | $F$-space (not LCS) | None (not locally convex) | Yes | Yes | [/example] ## Problems [problem] Let $X$ be a vector space and let $\mathcal{P} = \{p_\alpha\}_{\alpha \in A}$ be a separating family of seminorms on $X$. Show that the topology $\tau_\mathcal{P}$ generated by $\mathcal{P}$ makes $X$ into a Hausdorff locally convex TVS. Specifically: (a) Verify that addition $X \times X \to X$ is continuous. (b) Verify that scalar multiplication $\mathbb{R} \times X \to X$ is continuous. (c) Show that $\tau_\mathcal{P}$ is Hausdorff if and only if $\mathcal{P}$ is separating. [/problem] [solution] **Step 1: Continuity of addition.** We must show that for every basic neighbourhood $U = U_{\alpha_1, \dots, \alpha_n; \varepsilon}$ of $0$ and every $x_0, y_0 \in X$, there exist neighbourhoods $V$ of $x_0$ and $W$ of $y_0$ with $V + W \subseteq (x_0 + y_0) + U$. Take $V = x_0 + U_{\alpha_1, \dots, \alpha_n; \varepsilon/2}$ and $W = y_0 + U_{\alpha_1, \dots, \alpha_n; \varepsilon/2}$. If $v \in V$ and $w \in W$, write $v = x_0 + v'$ and $w = y_0 + w'$ with $p_{\alpha_k}(v') < \varepsilon/2$ and $p_{\alpha_k}(w') < \varepsilon/2$ for each $k$. Then $v + w = (x_0 + y_0) + (v' + w')$ and \begin{align*} p_{\alpha_k}(v' + w') \le p_{\alpha_k}(v') + p_{\alpha_k}(w') < \varepsilon \end{align*} by the triangle inequality, so $v + w \in (x_0 + y_0) + U$. **Step 2: Continuity of scalar multiplication.** We must show that for every basic neighbourhood $U = U_{\alpha_1, \dots, \alpha_n; \varepsilon}$ of $0$, every $\lambda_0 \in \mathbb{R}$, and every $x_0 \in X$, there exist $\delta > 0$ and a neighbourhood $V$ of $x_0$ such that $\lambda V \subseteq \lambda_0 x_0 + U$ for all $|\lambda - \lambda_0| < \delta$. Set $R := \max_k p_{\alpha_k}(x_0) + 1$ and choose $\delta := \min\{1, \varepsilon/(2R)\}$ and $V := x_0 + U_{\alpha_1, \dots, \alpha_n; \varepsilon/(2(|\lambda_0| + 1))}$. For $|\lambda - \lambda_0| < \delta$ and $x = x_0 + x' \in V$: \begin{align*} \lambda x - \lambda_0 x_0 = \lambda_0 x' + (\lambda - \lambda_0)x_0 + (\lambda - \lambda_0)x'. \end{align*} Estimating each seminorm $p_{\alpha_k}$: $p_{\alpha_k}(\lambda_0 x') \le |\lambda_0| \cdot p_{\alpha_k}(x') < |\lambda_0| \cdot \varepsilon/(2(|\lambda_0|+1)) \le \varepsilon/2$. And $p_{\alpha_k}((\lambda - \lambda_0)(x_0 + x')) \le \delta(p_{\alpha_k}(x_0) + p_{\alpha_k}(x')) \le \delta \cdot R < \varepsilon/2$. Therefore $p_{\alpha_k}(\lambda x - \lambda_0 x_0) < \varepsilon$, so $\lambda x \in \lambda_0 x_0 + U$. **Step 3: Hausdorff if and only if separating.** If $\mathcal{P}$ is separating and $x \neq y$, there exists $\alpha$ with $p_\alpha(x - y) > 0$. Set $\varepsilon := p_\alpha(x - y)/2$. Then $x + U_{\alpha; \varepsilon}$ and $y + U_{\alpha; \varepsilon}$ are disjoint: if $z$ belonged to both, then $p_\alpha(x - y) \le p_\alpha(x - z) + p_\alpha(z - y) < 2\varepsilon = p_\alpha(x - y)$, a contradiction. Conversely, if $\mathcal{P}$ is not separating, there exists $x \neq 0$ with $p_\alpha(x) = 0$ for all $\alpha$. Then $x$ belongs to every basic neighbourhood of $0$, so $0$ and $x$ cannot be separated. [/solution] [problem] Show that $C^\infty([0,1])$ (smooth functions on the closed interval) is a Fréchet space. Specifically, define the seminorms $p_m(f) := \sup_{x \in [0,1]} |f^{(m)}(x)|$ for $m \in \mathbb{N}_0$, and: (a) Verify that $\{p_m\}_{m=0}^\infty$ is a countable separating family of seminorms. (b) Show that the resulting metric space is complete: every Cauchy sequence converges in $C^\infty([0,1])$. [/problem] [solution] **Step 1: Countable and separating.** The family $\{p_m\}_{m=0}^\infty$ is countable (indexed by $m \in \mathbb{N}_0$). It is separating: if $p_m(f) = 0$ for all $m$, then in particular $p_0(f) = \sup_{x \in [0,1]} |f(x)| = 0$, so $f = 0$. **Step 2: Completeness.** Let $\{f_j\}_{j=1}^\infty$ be a Cauchy sequence in $C^\infty([0,1])$: for every $m$ and every $\varepsilon > 0$, there exists $J$ such that $p_m(f_j - f_k) = \sup_{x} |f_j^{(m)}(x) - f_k^{(m)}(x)| < \varepsilon$ for all $j, k \ge J$. This means $\{f_j^{(m)}\}$ is uniformly Cauchy on $[0,1]$ for each $m$. For $m = 0$: by the completeness of $C([0,1])$ with the sup norm, $f_j \to g_0$ uniformly for some $g_0 \in C([0,1])$. For $m = 1$: $\{f_j'\}$ [converges uniformly](/page/Uniform%20Convergence) to some $g_1 \in C([0,1])$. Since $f_j \to g_0$ uniformly and $f_j' \to g_1$ uniformly, the standard theorem on [interchange of limits and derivatives](/theorems/260) gives $g_0 \in C^1([0,1])$ with $g_0' = g_1$. Proceeding by induction: for each $m$, $\{f_j^{(m)}\}$ converges uniformly to some $g_m \in C([0,1])$, and $g_{m-1}' = g_m$. Therefore $g_0 \in C^\infty([0,1])$ with $g_0^{(m)} = g_m$ for every $m$. Setting $f := g_0$: $p_m(f_j - f) = \sup_x |f_j^{(m)}(x) - g_m(x)| \to 0$ for every $m$, so $f_j \to f$ in $C^\infty([0,1])$. [/solution] ## References - Rudin, *Functional Analysis* (1991), Chapters 1–3. - Schaefer and Wolff, *Topological Vector Spaces* (1999), Chapters I–IV. - Trèves, *Topological Vector Spaces, Distributions and Kernels* (1967), Chapters 1–14. - Conway, *A Course in Functional Analysis* (1990), Chapter V. - Brezis, *Functional Analysis, [Sobolev Spaces](/page/Sobolev%20Space) and Partial Differential Equations* (2011), Appendix.