Banach Spaces

Definition

A Banach space is a complete normed vector space.

That is, a vector space $V$ over $\mathbb{R}$ or $\mathbb{C}$ equipped with a norm $\Vert \cdot \Vert :V\to [0,\infty )$ such that every Cauchy sequence converges.

Examples

• ${\mathbb{R}}^n$ with any ${\ell }^p$ norm
• $L^p(\mu )$ for $1\le p\le \infty$
• $C([0,1])$ with the supremum norm

Key Results

Open Mapping Theorem

A surjective bounded linear operator between Banach spaces is an open map.

Closed Graph Theorem

A linear operator $T:X\to Y$ between Banach spaces is bounded if and only if its graph is closed.

Hahn-Banach

Every bounded linear functional on a subspace extends to the whole space.