----- > [!proposition] Proposition. ([[Mazur's Lemma]]) > Let $X$ be a [[norm|normed]] [[vector space]] and suppose $C$ is a nonempty [[convex set|convex subset]] of $X$ (e.g., a [[linear subspace]]). Then the [[weak topology|weak]] and [[norm]]-[[closure|closures]] are equal: $\overline{C}^{\|\cdot\|}=\overline{C}^{\sigma(X, X^{*})}.$ ^proposition > [!proposition] Corollary. > If $(x_{n})_{n=1}^{\infty}$ is a [[sequence]] in a [[norm|normed vector space]] [[converge|converging]] [[initial topology|weakly]] to $x \in X$, then there is a [[sequence]] of (finite) [[convex combination|convex combinations]] of the $x_{n}$ [[converge|converging]] in norm to $x$. ^proposition-2 > [!proof] Proof of Corollary. > The collection of all convex combinations of the $x_{n}$ forms the [[convex hull]] $C=\operatorname{ConvHull}(\{ x_{n} \}_{n=1}^{\infty}) \subset X$. Suppose $(x_{n}) \to x$ weakly to $x \in X$. Obviously $(x_{n}) \subset C$, thus by [[the sequence lemma]] $x \in \overline{C}^{\sigma (X, X^{*})}$, which we now know equals $\overline{C}^{\|\cdot\|}$, and so (again by [[the sequence lemma]]) there exists a sequence of elements in $C$ converging in norm to $x$. > ![[the sequence lemma#^181237]] ^proof-of-corollary > [!basicexample] Example of the Corollary. > [[Lp-norm|Put]] $X=\ell^{p}$ with $1<p<\infty$ and let $e_{n}$ be the standard vector with $1$ in the $n$th position. > > First note $(e_{n}) \to 0$ weakly. Indeed, since $1<p<\infty$, per [[Lp duality]] the map $\begin{align} > \ell^{p'} &\to (\ell^{p})^{*} \\ > \boldsymbol y & \mapsto \sum_{n=1}^{\infty} (\cdot)_{n} y _{n} > \end{align}$ > is an [[isomorphism]]. Thus given $f \in (\ell^{p})^{*}$, there exists $\boldsymbol y=(y_{n}) \in \ell^{p'}$ such that $f(\boldsymbol x)=\sum_{n=1}^{\infty} x_{n}y_{n} \text{ for all } \boldsymbol x \in \ell^{p}.$ > Thus $f(e_{n})=y_{n}$, and $\lim_{n \to \infty}f(e_{n})=\lim_{n \to \infty}y_{n}=0$ (since $\boldsymbol y$ is $p'$-summable). The sequence $(e_{n})$ does not converge in norm at all, however, the sequence $\boldsymbol g=(g_{m}) \subset \ell^{p}$ of convex combinations defined as $g_{m}=\frac{1}{m}\sum_{k=1}^{m} e_{k}, \text{ so } g_{m}(n)=\frac{1}{m}.$converges in $p$-norm to $0$. Indeed, $\|g_{m}\|_{p}^{p}=\sum_{n=1}^{\infty} |g_{m}(n)|^{p}=\sum_{n=1}^{m} \frac{1}{m^{p}}=\frac{m}{m^{p}} \to 0\text{ as } m \to \infty.$ > (Notice that $p=1$ would fail above.) ^example-of-corollary > [!proof]- Proof. ([[Mazur's Lemma]]) > Assume $\mathbb{F}=\mathbb{R}$ WLOG. > $\subset$. This is clear from the fact that $\sigma(X,X^{*})$ is [[comparable topologies|coarser]] than $\tau_{\|\cdot\|}$: the intersection of all $\tau_{\|\cdot\|}$-closed sets containing $C$ equals the intersection of all $\sigma(X,X^{*})$-closed sets containing $C$ together with potentially additional sets. > > $\supset$. Suppose $x_{0} \not \in \overline{C}^{\|\cdot\|}$. Then since $\overline{C}^{\|\cdot\|}$ is [[convex set|convex]] (closure in norm topology preserves convexity), the [[Hahn-Banach Separation theorem]] applies to give that $x_{0}$ and $\overline{C}^{\|\cdot\|}$ are [[linearly separable|uniformly separated]]: there is $f \in X^{*}$ and $\tau>0$ such that $f(x_{0}) >\tau > \sup_{x \in C} \{ f(x)\}=:\alpha$ . Put $\varepsilon:=\tau- \alpha>0$. Consider the basic weak neighborhood $U=\{ x \in X: |f(x-x_{0})|< \varepsilon \}$. Now, if $x \in U$, we have $f(x)=f(x_{0})+f(x-x_{0})$, hence $f(x) \geq f(x_{0})- |f(x-x_{0})|$, thus $f(x) \geq f(x_{0}) - |f(x-x_{0})| > \tau - \varepsilon = \alpha.$On the other hand, if $x \in C$ then $f(x) \leq \alpha$ by construction of $f$. So $U \cap C=\emptyset$. By [[neighborhood-basis characterization of set closure]], we're done. > > The case $\mathbb{F}=\mathbb{C}$ follows by wrapping with $\operatorname{Re}(\cdot)$ in the above argument. > ^28580b ![[CleanShot 2025-11-15 at [email protected]]] ----- #### ---- #### 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 > ```