---- [[field|Let]] $\mathbb{F}$ denote $\mathbb{R}$ or $\mathbb{C}$. $X$ and $Y$ are $\mathbb{F}$-[[vector space|vector spaces]]. > [!definition] Definition. ([[weak topology]]) > > Suppose $(X,Y, \langle -,- \rangle)$ is a [[orthogonal complement|dual pair]].[^1] > > - The **weak topology on $X$** is the [[initial topology]] with respect to the collection $\{ \langle -,y \rangle: y \in Y \}$ of [[dual vector space|linear functionals]] $X \to \mathbb{F}$. It is denoted $\sigma(X,Y)$. > - The **weak topology on $Y$** is the [[initial topology]] with respect to the collection $\{ \langle x,- \rangle : x \in X \}$ of [[dual vector space|linear functionals]] $Y\to \mathbb{F}$. It is denoted $\sigma(Y,X)$. > > In other words, the weak topology on $X$ resp. $Y$ is the [[comparable topologies|coarsest]] [[topological space|topology]] such that $X$ resp. $Y$ is a [[topological vector space|TVS]] wherein $\langle x \in X, - \rangle$ resp. $\langle -, y \in Y \rangle$ are [[continuous]]. > [!equivalence] Local convexity equivalence. > Being an [[initial topology]], $\sigma(X,Y)$ is generated by the [[subbasis for a topology|subbasis]] $\mathscr{S}=\{ \langle -,y \rangle^{-1} (B_{r}(z)): z \in \mathbb{F},r>0, y \in Y\},$ > where $\langle -,y \rangle ^{-1}\big( B_{r}(z) \big)= \{ x \in X: |\langle x,y \rangle - z | < r \}.$ > Define $p_{y}:= |\langle -,y \rangle|$. Since $\langle -, y \rangle$ is not zero, it is [[surjection|surjective]], meaning that for every $z \in \mathbb{F}$ there exists $x_{0} \in X$ with $\langle x_{0},y \rangle=z$. Then $\langle -,y \rangle ^{-1}\big( B_{r}(z) \big)=x_{0} + \{ x \in X : p_{y}(x)< r \}.$ > Thus the subbasic sets are translates of the $0$-neighborhoods $\{ x: p_{y}(x)< r \}$. Taking finite intersections gives a [[first-countable space|neighborhood basis]] of $0$; explicitly, elements of $\mathscr{B}_0$ are of the form $B_{(\varepsilon_{k})_{k=1}^{n}}\big(0; y_{1},\dots,y_{n} \big)= \{ x \in X : p_{y_{k}}(x) < \varepsilon_{k} \text{ for all } k=1,\dots,n \}$ > and translating [[condition for obtaining a basis from a topology|gives]] a [[basis for a topology|basis]] [[topology generated by a basis|generating]] $\tau_{\text{weak}}$. > > Note that the maps $\{ p_{y} \}_{y \in Y}$ are [[seminorm|seminorms]]. Consequently, the weak topology makes $X$ into a [[locally convex space]]. Moreover, since [[orthogonal complement|dual pairs]] by definition separate points, $(X, \tau_{\text{weak}})$ is [[Hausdorff space|Hausdorff]]. > > > (The same all holds for $\sigma(Y,X)$ by switching roles of $X,Y$.) > [!specialization] Specializing. (Weak and weak-star topologies) > The most important special case is when $X$ is a [[vector space]], $X^{*}$ is its algebraic [[dual vector space|dual space]], and $\langle -,- \rangle:X^{*} \times X \to \mathbb{F}$ is the natural pairing $\langle f, x \rangle:= f(x)$. (If $X$ is already a [[topological vector space|TVS]] then $X^{*}$ should be the [[continuous]] dual.) > In this case, the weak topology $\sigma(X, X^{*})$ on $X$ is generated by the (translates of) the basic open sets $B_{(\varepsilon_{k})_{k=1}^{n}}(0; f_{1},\dots,f_{n})=\{ x \in X: |f_{k}(x)|< \varepsilon_{k} \text{ for all }k=1,\dots,n \},$while the weak topology $\sigma(X^{*}, X)$ on $X^{*}$ (usually called the **weak-$*$ topology**) is generated by translates of the basic $0$-neighborhoods $B_{(\varepsilon_{k})_{k=1}^{n}}(0; x_{1},\dots,x_{n})= \{ f \in X^{*}: |f(x_{k})|<\varepsilon_{k} \text{ for all }k=1,\dots,n \}.$ > By definition, $\sigma(X,X^{*})$ is the coarsest topology on $X$ for which elements of $X^{*}$ are all [[continuous]]. Since [[initial topology|initial topology is topology of pointwise convergence]], a [[sequence]] $(x_{n})$ converges weakly to $x \in X$ iff $f_{n}(x) \to f(x)$ for all $f \in X^{*}$. > Meanwhile... same thing for $\sigma(X^{*}, X)$. (Note that can actually just choose a single $\varepsilon>0$) > [!basicexample] > - $\mathbb{B}_{X}$ is weakly closed. $\mathbb{B}_{X^{*}}$ is weak-$*$ closed. ^basic-example > [!basicnonexample] > > - $\mathbb{S}_{X}$ is weakly closed iff $\text{dim }X<\infty$. Similarly, $\mathbb{S}_{X^{*}}$ is weak$*$ closed iff $\text{dim }X^{*}<\infty$. > - $\mathbb{B}_{X}^{\circ}$ is weakly open iff $\text{dim }<\infty$. Similarly, $\mathbb{B}_{X^{*}}^{\circ}$ weak$*$ open iff $\text{dim }X^{*}<\infty$. > [!proof]- > One direction is clear since weak topology=norm topology in finite dimensions. For the other, assume $\text{dim }X=\infty$. We claim $0 \in \overline{\mathbb{S}_{X}}^{\sigma(X, X^{*})}$. [[neighborhood-basis characterization of set closure|Indeed]], let $\varepsilon>0$ and $f_{1},\dots.,f_{k} \in X^{*}$ and consider the basic open set $U_{f_{1},\dots,f_{k}, \varepsilon}(0)=\{ x \in X: |f_{1}(x)|<\varepsilon, \dots, |f_{k}(x)|<\varepsilon \}.$ > We claim it intersects $\mathbb{S}_{X}$ nontrivially. Indeed, by properties in [[orthogonal complement|preannihilator]], we know $F_{\circ}:= \bigcap_{i=1}^{k}\operatorname{ker }f_{i} \neq (0)$ since $\text{dim }X=\infty$ (briefly: if it were $(0)$, then $T=(f_{1},\dots,f_{k}):X \to \mathbb{F}^{k}$ would be injective, witnessing $X$ to be finite-dimensional). So let nonzero $x \in \bigcap_{i=1}^{k} \operatorname{ker }f_{i}$, which we can normalize without leaving $U$ so assume $\|x\|=1$. Now, $x \in \mathbb{S}_{X}$, but also $|f_{1}(x)|=0<\varepsilon$, ..., $|f_{k}(x)|=0<\varepsilon$, thus $f \in U_{f_{1},\dots,f_{k}, \varepsilon}(0)$. This finishes. > > **Weak-$*$ sphere.** Similarly, claim the zero functional is in $\overline{\mathbb{S}_{X^{*}}}^{\sigma(X^{*}, X)}$. Indeed, take any weak$*$ basic open neighborhood $U=U_{x_{1},\dots,x_{n}}(0)$. [[reflexive space|Dualize]] the $x_{i}$ to $\varphi_{i}=\_(x_{i}) \in X^{* *}$. Now $\bigcap_{i=1}^{n} \operatorname{ker } \varphi_{i} \neq (0)$, since if it where the map $f \mapsto \big( \varphi_{1}(f), \dots, \varphi_{n}(f) \big)$ would inject $X^{*}$ into $\mathbb{F}^{k}$, contradicting $\text{dim }=\infty$. Let $0 \neq f \in \bigcap_{i=1}^{n} \operatorname{ker } \varphi_{i}$. We have $f \in U$, since $\varphi_{i}(f)=0 \iff f(x_{i})=0$, and so also $\frac{f}{\|f\|} \in U \cap \mathbb{S}_{X^{*}}$ > > **Ball.** The argument for $\mathbb{B}_{X}^{\circ}$ is along the same lines. Claim that when $\text{dim }X=\infty$, there does not exist a weak basic $0 \in U \in \mathbb{B}_{X}^{\circ}$. Indeed, take any weak basic $U=U_{f_{1},\dots,f_{k}}(0)$ about $0$. Have $\bigcap_{i=1}^{k} \operatorname{ker }f_{i} \neq (0)$, since otherwise $X \hookrightarrow \mathbb{F}^{k}$ contradicting $\text{dim }X=\infty$. Let nonzero $x \in \bigcap_{i=1}^{k} \operatorname{ker } f_{i}$; then $\frac{x}{\|x\|}$ is again in $U$ but not $\mathbb{B}_{X}^{\circ}$. > > **Weak ball.** You know the drill. Note also that a function $F:(Z, \tau) \to (X, \sigma(X, X^{*}))$ is weak-continuous iff $f \circ F:Z \to \mathbb{R}$ is $\tau$-continuous for all $f \in X^{*}$. > [!note] > On $X^{*}$ we default to considering the weak-$*$ topology $\sigma(X^{*},X)$, but would also consider the weak topology $\sigma(X^{* *},X^{*})$. The two agree if and only if $X$ is [[reflexive space|reflexive]], as follows from [[there are no weakly continuous functionals past the generating set|the discussion here]]. ^note - [ ] norm weakly continuous iff in finite dimensions (but always [[weakly lower semicontinuous|WLSC]]); point is that in infinite dimensions that open ball won't be weakly open ---- #### [^1]: That is, $X,Y$ are $\mathbb{F}$-[[vector space|vector spaces]] and $\langle -,- \rangle:X \times Y \to \mathbb{F}$ is a [[bilinear map|bilinear form]] that [[weak topology|separates points]] ($\langle x,- \rangle$ and $\langle -,y \rangle$ are [[injection|injective]] for all $x \in X$ and $y \in Y$). ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```