Banach space

A Banach space $(X,\norm{\,\cdot\,})$ is a normed vector space such that $X$ is complete under the metric induced by the norm $\norm{\,\cdot\,}$ .

Some authors use the term Banach space only in the case where $X$ is infinite-dimensional, although on Planetmath finite-dimensional spaces are also considered to be Banach spaces.

If $Y$ is a Banach space and $X$ is any normed vector space, then the set of continuous linear maps $f\colon X\to Y$ forms a Banach space, with norm given by the operator norm. In particular, since $\mathbb{R}$ and $\mathbb{C}$ are complete, the continuous linear functionals on a normed vector space $\mathcal{B}$ form a Banach space, known as the dual space of $\mathcal{B}$ .

Examples:

• Finite-dimensional normed vector spaces.
• $L^p$ spaces are by far the most common example of Banach spaces.
• $\ell^p$ spaces are $L^p$ spaces for the counting measure on $\mathbb{N}$ .
• Continuous functions on a compact set under the supremum norm.
• Finite signed measures on a $\sigma$ -algebra.