# Banach space

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

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 (http://planetmath.org/EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace).

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

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

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

Title Banach space BanachSpace 2013-03-22 12:13:48 2013-03-22 12:13:48 bbukh (348) bbukh (348) 11 bbukh (348) Definition msc 46B99 msc 54E50 VectorNorm DualSpace dual space