Banach-Alaoglu Theorem/Lemma 1

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Lemma for Banach-Alaoglu Theorem

Let $X$ be a separable normed vector space.

Let $X^*$ be the dual space of $X$.

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.

$l$ can be extended to an element of $X^*$.

Proof

$l$ can be extended in the obvious way to a linear function on $M = \span \set {x_j}_{j \mathop \in \N}$.

We extend it to a functional in $X^*$ by pointwise limit (notice that $M$ is dense in $X$).

We have:

$\ds \size {\map l x} = \lim_{k \mathop \to \infty} \size {\map l {x_k} } \le \limsup_{k \mathop \to \infty} \norm {l_k}_{X^*} \norm x_X$

where $x_k \to x$ as $k \to \infty$.

Since $\set {l_k}_k \mathop \in \N$ was bounded, $l$ is bounded and thus continuous.

$\blacksquare$