Banach-Alaoglu Theorem
Theorem
Let $X$ be a separable normed vector space.
Then the closed unit ball in its dual $X^*$ is weak* sequentially compact.
Proof 1
The aim of this proof is to show the following:
Given a bounded sequence in $X^*$, there exists a weakly convergent subsequence of that bounded sequence.
Let $\sequence {l_n}_{n \mathop \in \N}$ be a bounded sequence in $X^*$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a countable dense subset of $X$.
Choose subsequences $\N \supset \Lambda_1 \supset \Lambda_2 \supset \ldots$ such that:
- $\forall j \in \N: \map {l_k} {x_j} \to a_j =: \map l {x_j}$
as $k \to \infty$, $k \in \Lambda_j$.
Let $\Lambda$ be the diagonal sequence.
Lemma 1
$l$ can be extended to an element of $X^*$.
$\Box$
Lemma 2
$l_k \stackrel {\omega^*} {\to} l$ as $k \to \infty$, $k \in \Lambda$.
$\Box$
Proof 2
Let $X$ be a normed vector space.
Denote by $B$ the closed unit ball in $X$.
Let $X^*$ be the dual of $X$.
Denote by $B^*$ the closed unit ball in $X^*$.
Let $\map \FF B = \closedint {-1} 1^B$ be the topological space of functions from $B$ to $\closedint {-1} 1$.
By Tychonoff's Theorem, $\map \FF B$ is compact with respect to the product topology.
We define the restriction map:
- $R: B^* \to \map \FF B$
by:
- $\map R \psi = \psi \restriction_B$
Lemma 3
$\map R {B^*}$ is a closed subset of $\map \FF B$.
$\Box$
Lemma 4
$R$ is a homeomorphism from $B^*$ with the weak* topology to its image $\map R {B^*}$ seen as a subset of $\map \FF B$ with the product topology.
$\Box$
Thus by Lemma 4, $B^*$ in the weak* topology is homeomorphic with $\map R {B^*}$.
This is a closed subset of $\map \FF B$ (by Lemma 3) and thus compact.
$\blacksquare$
Also known as
The Banach-Alaoglu Theorem is also known just as Alaoglu's Theorem.
Source of Name
This entry was named for Stefan Banach and Leonidas Alaoglu.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Alaoglu's theorem
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Banach-Alaoglu theorem
- 2010: H.L. Royden and P.M. Fitzpatrick: Real Analysis (4th ed.): $\S 15.1$
- 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control: $3.3$: The weak* topology