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Work over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. A Fréchet space is a Hausdorff, complete, metrizable locally convex space. Its topology is generated by a countable separating family of seminorms $(p_n{)}_{n\ge 1}$. A neighbourhood base at zero consists of finite intersectionsReplacing $p_n$ by ${\max }_{j\le n}{p}_j$ makes the family increasing. One compatible translation-invariant metric isCompleteness means that a sequence Cauchy for all these seminorms converges in the space. The separating condition ensures that $d(x,y)=0$ only when $x=y$.

Let $E^{\prime }$ denote the continuous dual space for the original topology $\tau$. The weak topology $\sigma (E,E^{\prime })$ is the coarsest topology making all $f\in E^{\prime }$ continuous. Its basic zero-neighbourhoods areEquivalently, it is generated by the seminorms $x\mapsto |f(x)|$.

To prove the Hausdorff property, fix $z\ne 0$. Some continuous seminorm $p$ has $p(z)>0$. On the one-dimensional span of $z$, define $f_0(az)=ap(z)$. Then $|f_0(az)|=p(az)$. The real or complex Hahn-Banach theorem extends this to $f:E\to \mathbb{F}$ with $|f(x)|\le p(x)$. Thus $f$ is originally continuous and $f(z)\ne 0$. This proves continuous-dual point separation. For distinct $x,y$, inverse images under such an $f$ of disjoint small scalar neighbourhoods of $f(x)$ and $f(y)$ are disjoint weak neighbourhoods. Hence the weak topology is Hausdorff.

If a linear functional is originally continuous, it belongs to $E^{\prime }$ and is weakly continuous by definition. Conversely, $\sigma (E,E^{\prime })$ is coarser than $\tau$, so every weakly continuous scalar function is originally continuous. In particular,This special case of the continuous dual of a weak topology needs no completeness argument.

For the linear map $T:E_1\to E_2$, original continuity implies that $g\circ T\in E_1^{\prime }$ whenever $g\in E_2^{\prime }$. Each such composition is weakly continuous on $E_1$, so the defining property of the target weak topology makes $T$ weak-to-weak continuous.

Conversely, suppose $T$ is weak-to-weak continuous. Every $g\circ T$, $g\in E_2^{\prime }$, is then weakly continuous, and hence continuous for ${\tau }_1$. We show that the graph of a linear operator $\Gamma (T)$ is closed in the original product topology. If $x_n\to x$ in $E_1$ and $T{x}_n\to y$ in $E_2$, thenBy continuous-dual point separation on $E_2$, this forces $y=Tx$. The product is metrizable, so sequential closedness is closedness. The closed graph theorem for Fréchet spaces now gives original continuity of $T$.

The relevant version of that theorem does use completeness of both spaces: a closed graph is a Fréchet space, its first-coordinate projection is a continuous linear bijection onto $E_1$, and the open mapping theorem for Fréchet spaces gives a continuous inverse. Composing with the second-coordinate projection gives $T$. ThusEquality of continuous scalar functionals does not imply equality of the two topologies; the closed graph theorem for Fréchet spaces is the extra step that makes this conclusion valid.

A separated dual pair consists of real vector spaces $E,F$ and a bilinear form $⟨x,y⟩$ that separates both variables: each nonzero $x\in E$ pairs nontrivially with some $y\in F$, and conversely. Thus $F$ embeds in the algebraic dual of $E$, and $E$ embeds in the algebraic dual of $F$.

The weak topology $\sigma (E,F)$ is the coarsest topology making all maps $x\mapsto ⟨x,y⟩$, $y\in F$, continuous. It makes $E$ a Hausdorff locally convex space, with topology generated by the seminorms $p_y(x)=|⟨x,y⟩|$. A neighbourhood base at zero consists of finite intersectionsSeparation ensures that the common zero set of all these seminorms is just zero, giving Hausdorffness.

We determine the continuous dual of a weak topology. Suppose a linear functional $\ell$ is continuous. A basic neighbourhood as above is contained in $\{x:|\ell (x)|<1\}$. If $x$ is in the common kernel of the finitely many evaluations, every scalar multiple of $x$ lies in that neighbourhood, so $\ell (x)=0$. Hence $\ell$ factors through the image ofExtend the resulting linear functional on $T(E)$ to ${\mathbb{R}}^m$. It has the form $z\mapsto {\sum }_j{a}_j{z}_j$, giving $\ell (x)=⟨x,{\sum }_j{a}_j{y}_j⟩$. Conversely every such evaluation is continuous by the definition of the weak topology. Since the pairing separates $F$, this identification is injective, andFor the remaining parts every topological assertion about $E^{\prime }$ or its subsets uses the stipulated weak-star topology $\sigma (E^{\prime },E)$.