Babuška–Lax–Milgram theorem

In functional analysis, a branch of mathematics, the Shilov boundary is the smallest closed subset of the structure space of a commutative Banach algebra where an analog of the maximum modulus principle holds. It is named after its discoverer, Georgii Evgen'evich Shilov.

Precise definition and existence

Let $\mathcal{𝒜}$ be a commutative Banach algebra and let $\mathrm{\Delta }\mathcal{𝒜}$ be its structure space equipped with the relative weak*-topology of the dual ${\mathcal{𝒜}}^{*}$. A closed (in this topology) subset $F$ of $\mathrm{\Delta }\mathcal{𝒜}$ is called a boundary of $\mathcal{𝒜}$ if ${\max }_{f\in \mathrm{\Delta }\mathcal{𝒜}}|x(f)|={\max }_{f\in F}|x(f)|$ for all $x\in \mathcal{𝒜}$. The set $S=\bigcap \{F:F\text{ is a boundary of }\mathcal{𝒜}\}$ is called the Shilov boundary. It has been proved by Shilov that $S$ is a boundary of $\mathcal{𝒜}$.

Thus one may also say that Shilov boundary is the unique set $S\subset \mathrm{\Delta }\mathcal{𝒜}$ which satisfies

1. $S$ is a boundary of $\mathcal{𝒜}$, and
2. whenever $F$ is a boundary of $\mathcal{𝒜}$, then $S\subset F$.

Examples

• Let $\mathbb{𝔻}=\{z\in \mathbb{ℂ}:|z|<1\}$ be the open unit disc in the complex plane and let

$\mathcal{𝒜}=\mathcal{\mathscr{H}}(\mathbb{𝔻})\cap \mathcal{𝒞}(\overline{\mathbb{𝔻}})$ be the disc algebra, i.e. the functions holomorphic in $\mathbb{𝔻}$ and continuous in the closure of $\mathbb{𝔻}$ with supremum norm and usual algebraic operations. Then $\mathrm{\Delta }\mathcal{𝒜}=\overline{\mathbb{𝔻}}$ and $S=\{|z|=1\}$.

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See also

• James boundary