Banach space

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

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 $f:X\to Y$ forms a Banach space, with norm given by the operator norm. In particular, since $ℝ$ and $ℂ$ are complete, the continuous linear functionals on a normed vector space $ℬ$ form a Banach space, known as the dual space of $ℬ$.

Examples:

• Finite-dimensional normed vector spaces (http://planetmath.org/EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace).

• $L^p$ spaces (http://planetmath.org/LpSpace) are by far the most common example of Banach spaces.

• ${\mathrm{\ell }}^p$ spaces (http://planetmath.org/Lp) are $L^p$ spaces for the counting measure on $\mathbb{N}$.

• Continuous functions on a compact set under the supremum norm.

• Finite (http://planetmath.org/FiniteMeasureSpace) signed measures on a $\sigma$-algebra (http://planetmath.org/SigmaAlgebra).